Smart Braid Feedback for the Closed-Loop Control of Soft Robotic Systems

Estimation of revolute joint angle with Smart Braids

The actuator neutral lengths \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{neutral}}$$ \end{document} were defined as the lengths of the actuators when the joint angle was 0° and the connections to the joint were without slack, but unstretched. The actuator contractions x were the deviations of the actuator lengths l away from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{neutral}}$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{x_1} = {l_{1 , neutral}} - {l_1}{ \kern 1pt} } \hfill \\ {{x_2} = {l_2} - {l_{2 , neutral}}.} \hfill \\ \end{matrix} \tag{1} \end{align*} \end{document}

For each actuator, a linear function of the inductance L was used to estimate the actuator contraction \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$x ( L )$$ \end{document} ⁴⁰ \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{x_1} ( {L_1} ) = {a_{r1}}{L_1} + {b_{r1}}{ \kern 1pt} } \hfill \\ {{x_2} ( {L_2} ) = {a_{r2}}{L_2} + {b_{r2}}.} \hfill \\ \end{matrix} \tag{2} \end{align*} \end{document}

If the connections between the actuator and joint were perfectly stiff, the length of the actuators could be used to directly determine the joint angle. Our prior work, however, suggests that measurements of joint motion can be skewed by compliance in mechanical couplings between the joint and actuator.^39,51 We thus compensated for this compliance by modeling the force output of the actuator and the compliance of the connection points. This compliance compensation used measurements of the actuator pressure to estimate the actuator force output. With the assumption that the connections were under tension, the displacement of the actuators was written in terms of the joint angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} and the output force F. This relationship took into account the stretching of the actuator connections (with stiffness k) \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{split} { { x_1 } } { = r \alpha + { \frac { { F_1 } } { { k_1 } } } } \hfill \\ { { x_2 } } { = r \alpha - { \frac { { F_2 } } { { k_2 } } } . } \hfill \\ \end{split} \tag { 3 } \end{align*} \end{document}

The estimates of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} that result from inverting these relationships will be most accurate when the forces F are small and the stiffnesses k of the connections are high.

Calibration of the revolute joint

To characterize the force, pressure, and contraction relationship of the actuators, an empirical function was used. It was based on the contraction ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \epsilon _i}$$ \end{document} , ⁵² which is the normalized difference between the fully extended actuator length l_e and the current actuator length l_i \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \epsilon _i } = { \frac { { l_e } - { l_i } } { { l_e } } } . \tag { 4 } \end{align*} \end{document}

The estimated actuator force output \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \hat F_i}}$$ \end{document} was modeled as a polynomial that is linear with respect to actuator pressure P_i and quadratic with respect to the contraction ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \epsilon _i}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { {\hat F_i}} = {p_{00}} + {p_{10}}{P_i} + {p_{01}}{ \epsilon _i} + {p_{11}}{P_i}{ \epsilon _i} + {p_{02}} \epsilon _i^2. \tag{5} \end{align*} \end{document}

To collect the necessary data for the calibration, the actuators were tested under a range of cable tensions and actuator pressure values. The tensions ranged between 5 and 30 N (with 5 N increments). At each tension level, the pressure of actuator 1 was set to each of a series of pressure values between 0.1 and .31 MPa for 30 s (in .035 MPa increments). During each 30 s period, the pressure in actuator 2 was adjusted automatically to create the desired tension. This adjustment process was driven by measurements of cable tension from the force transducer in line with actuator 2 (Fig. 2). In this way, data were collected at each combination of tension levels and actuator 1 pressures. After each combination of pressure and tension was tested on actuator 1, the process was repeated with actuator 2 at the fixed pressure values. In this case, the pressure in actuator 1 was adjusted to maintain the desired cable tension. The steady-state data from the last 15 s of each pressure–tension combination were used in the calibration.

The data from these experiments were used to identify the coefficients of Eq. (2) and the estimated connection stiffness \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\hat k$$ \end{document} in Eq. (3) (Table 1). The data were also used to identify the coefficients of Eq. (5) (Table 2). Note that connections to actuator 2 were less stiff than those to actuator 1. This was caused by the additional, compliant cable ends used to include the force transducer in the cable (Fig. 2). The low values of the coefficient of determination R² in Table 2 result, in part, from hysteresis effects, which are not modeled, ⁵² and the noisy measurements from the force transducer.

Compliance compensation and feedback control of the revolute joint

The goal of the feedback controller was to regulate the joint angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} to a desired angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \alpha _{des}}$$ \end{document} . Based on this desired angle, the relationships in Eq. (3) defined the desired length \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${x_{des}}$$ \end{document} of each actuator as a function of estimated actuator force outputs \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat F_i}$$ \end{document} and estimated connection stiffnesses \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat k_i}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{split} { x_ {1,des}} = r {\alpha _ {des} } + \frac {\hat F_1} { \hat k _1 } \\ {x_ {2,des } } = r { \alpha _ {des} } + \frac { \hat F_2 } { \hat k_2 } . \end{split} \tag { 6 } \end{align*} \end{document}

The force estimates \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat F_i}$$ \end{document} in the compensation terms were found by evaluating Eq. (5) with measurements of current pressure and actuator length. Length measurements were obtained from inductance values using Eq. (2).

The difference between the desired and measured contraction of each individual actuator constituted the control error e_i: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { e_i } = \frac { { { x_ { i , des } } - { x_i } \left( { { L_i } } \right) } } { r } . \tag { 7 } \end{align*} \end{document}

This error was normalized by the pulley radius r to yield values in units of joint angle. The complete compensation process is illustrated in Figure 5.

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FIG. 5.

The desired actuator contractions \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${x_{i , des}}$$ \end{document} included a compensation term to account for stretching in the actuator connections. This compensation term was based on estimates of the force output \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat F_i}$$ \end{document} and the stiffness of the connections \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat k_i}$$ \end{document} of each actuator. The control error between the desired and measured contraction was scaled by the radius r of the pulley to yield a joint angle error e in units of radians.

A separate PID controller for each actuator regulated the position errors e_i by commanding desired actuator forces \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${F_{i , des}}$$ \end{document} (Fig. 6). To maintain tension in both actuators, an equal “preload” force \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${F_{pre}}$$ \end{document} was added to each desired force value. Different levels of preload were evaluated experimentally. Because the actuators are unable to create negative forces, both \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${F_{i , des}}$$ \end{document} were saturated to be positive: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{split} & {{F_{1 , des}}} { = \max \left( {{k_p}{e_1} + {k_d}{{ \dot e}_1} + {k_i} {\displaystyle\int} {e_1} + {F_{pre}} , 0} \right) } \hfill \\ & {{F_{2 , des}}} { = \max \left( { - {k_p}{e_2} - {k_d}{{ \dot e}_2} - {k_i} {\displaystyle\int} {e_2} + {F_{pre}} , 0} \right) .} \hfill \\ \end{split} \tag{8} \end{align*} \end{document}

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FIG. 6.

For the revolute joint, two separate PID controllers commanded a desired force \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${F_{i , des}}$$ \end{document} for each actuator in reaction to estimated joint angle error terms e_i. When the Smart Braids were used for feedback, e_i were determined individually from the measured inductance and pressure. When the encoder was used for feedback, e_i were equal. To generate the desired forces, the controller computed desired pressure values from a model of the actuator force-pressure-length relationship and sent these values to two pressure control valves.

By inverting Eq. (5), the desired forces \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${F_{i , des}}$$ \end{document} were converted into desired pressure values for each actuator \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${P_{i , des}}$$ \end{document} and sent to pressure-control valves.

The performance of the Smart Braid feedback was compared to feedback from a rotary encoder on the joint. When using the encoder for feedback, the actuator-specific error value was simply the difference between the desired angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \alpha _{des}}$$ \end{document} and the measured angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \alpha _{measured}}$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {e_1} = {e_2} = { \alpha _{des}} - { \alpha _{measured}}. \tag{9} \end{align*} \end{document}

This error was used in both PID controllers of Eq. (8). No compliance compensation had to be performed.

Experimental implementation, revolute joint

A Texas Instruments “inductance-to-digital converter” (TI LDC1612/4) ⁵³ provided rapid measurements of the sensor inductance. This chip operates by exciting the natural frequency of a resonant tank circuit formed by an inductor and a capacitor in parallel. The Smart Braid sensors were connected in parallel with 390 pF capacitors. The series resistance of the Smart Braid sensors was ∼0.3 ohms. The inductance values from the sensors had a target sampling rate of 1 kHz. An analysis of the inductance measurements from the Smart Braids in relaxed conditions showed a root-mean-square (RMS) noise level of 0.24 nH.

In the revolute joint test fixture, a digital incremental encoder (Koyo Electronics Industries; TRDA-2E2500VD) provided joint angle measurements with a quadrature resolution of 0.036°. The pulley radius r where the cables were connected to the revolute joint was 25.4 mm. A force transducer (Omega LC201-100) was attached serially to the steel cable of actuator 2. This was used to characterize the force–pressure relationship of the actuators, characterize the stiffness of the actuator connections, and measure preload tracking performance. The systems were controlled with custom scripts in LabVIEW. The measurements from the test apparatus were collected and processed with a data acquisition unit (NI PXIe-6341), which used a PXI express chassis (NI PXIe-1073) to communicate with LabVIEW. Inductance measurements were collected using an I2C bus (NI USB-8451). The mass flow rate into the actuator lines was controlled with proportional pneumatic valves (Enfield LS-V05 s). The pressure in each actuator line was measured with pressure transducers (WIKA A-10) with a [0.41]MPa range and controlled with a custom controller with compensation for the nonlinear aperture flow across the valves. ⁵⁴ A 250 Hz LabVIEW loop acquired data from the pressure sensors and sent commands to the valves. The system pressure was limited to 0.31 MPa. Estimates of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\dot e$$ \end{document} were filtered with a five-point moving average. In the control loop, the measurements of inductance were filtered with a third-order low-pass Butterworth filter with a cutoff frequency of 10 Hz. To prevent excessive integrator windup, the product of the error integral and the integral gain was not permitted to exceed a magnitude of 50 N.

The two types of feedback were tested in random sequence at each of seven preload levels \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${F_{pre}}$$ \end{document} : 5 N, 7.5 N, 10 N, 12.5 N, 15 N, 17.5 N, and 20 N. Gains for each feedback type were selected by the “some-overshoot” Ziegler-Nichols method. ⁵⁵ We first conducted a set of tuning trials in which gains were identified individually at each preload level. From the identified gains, the most conservative ones were subsequently used in the controller evaluation, where they were kept identical for all commanded preload levels (Table 3). This was done to ensure control stability in the event that the preload level did not match the commanded level.

Each controller configuration was tested on a fixed sequence of step changes for the desired joint angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \alpha _{des}}$$ \end{document} . The sequence was a random series of 21 angles between −30° and 30°, which were held for 10 s each. The data from the first 5 s of each step were considered “transient” and the data from the last 5 s were considered “steady state.” The performance was quantified by the RMS of the absolute error between the reference angle ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \alpha _{des}}$$ \end{document} ) and the angle measured by the encoder. During these tests, no load torque was applied (Fig. 4a). Each feedback type was tested at a given preload level thrice. Statistical significance between the feedback types was determined by paired t-tests across each of the commanded angles in the combined three tests. The values of the measured preload were averaged over the last 5 s of each step in the sequence.

To evaluate the effectiveness of the compliance compensation in Eq. (6), the controller tests were repeated with the two load configurations that generated a nonzero net-torque (Fig. 3b, c). The unmodeled load torque renders open-loop control infeasible and stretches the actuator connections asymmetrically.

Results, revolute joint

Feedback control based entirely on soft, Smart Braid sensing is feasible. The Smart Braid feedback controller was able to track step changes in the commanded angle (Fig. 7). With no load torque, Smart Braid feedback led to an average RMS in the joint angle error of 1.73° (standard deviation [SD] 0.69°) during the last 5 s of each commanded angle (considered steady state, Fig. 8). The average RMS of the tracking error in the first 5 s of the commanded angles was 7.85° (SD 5.21°). Even with load torques of 0.65 Nm, the average steady-state error remained less than 2° (SD < 1°, Fig. 9).

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FIG. 7.

Comparison of the effect of feedback type on the reference tracking of the revolute joint controller. On average, the encoder feedback (red lines) resulted in better joint angle tracking than the Smart Braid feedback (blue lines). During the first 5 s after the commanded step change, the average RMS of the error was 16% smaller in the encoder feedback case. In the last 5 s, the RMS of the encoder feedback error was, on average, 51% smaller than the Smart Braid error. In the case shown in this study, the desired preload was 10 N and no load torque was applied. Reported values were measured with the rotary encoder. RMS, root mean square. Color images available online at www.liebertpub.com/soro

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FIG. 8.

Comparison of the steady-state error of the two feedback types used on the revolute joint under different preload conditions. Shown are the average RMS values across the three trials with no load torque. The vertical axis corresponds to the RMS of the reference error during the last 5 s of a commanded angle. The feedback from the encoder resulted in smaller errors in the steady-state tracking. Color images available online at www.liebertpub.com/soro

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FIG. 9.

Comparison of the steady-state error of the two feedback types used on the revolute joint under different load torque conditions. The height of the bars represents the averages of the RMS error across all the commanded preload levels during the last 5 s of each of each commanded angle. The average error with encoder feedback was always less than with inductance feedback. This difference was statistically significant (p < .05) in all cases except when \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \tau _{{ \rm{load}}}} =$$ \end{document} 0.65 Nm and the desired preload was 15 N or higher. Color images available online at www.liebertpub.com/soro

The positional accuracy of the inductance feedback controller was comparable to the performance of a controller with feedback from the rotary encoder. During the first 5 s after a step change in the reference angle, the average RMS of the tracking error with encoder feedback was 6.60° (16% lower than with Smart Braid feedback, SD 4.56°, Fig. 7). Encoder feedback resulted in steady-state (during the last 5 s) errors with average RMS values between 0.8° and 1.3° depending on the load torque (SD < 0.8°, Fig. 9). This was smaller than the 1.5°–1.8° average RMS errors exhibited with Smart Braid feedback.

Figure 10 shows the measured preloads averaged over the last 5 s of each reference angle in the controlled trials (no load torque). With Smart Braid feedback, preloads of nearly 70 N were observed in the 12 N and 20 N conditions. The large preloads resulted from integral windup in the physically antagonized, yet independent, PID controllers.

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FIG. 10.

Shown are the preloads on the revolute joint averaged over during the last 5 s of each commanded angle in the controlled trials. The two types of feedback resulted in different preloads. The effectual single integrator of the encoder feedback resulted in better preload tracking than the two integrators of the inductance feedback. For the conditions shown in this study, no load torque was applied. Color images available online at www.liebertpub.com/soro

Estimation of continuum manipulator tip angle with Smart Braids

The continuum manipulator was modeled as having a constant curvature enclosing an angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} . The length of the center of the manipulator was constant, constrained by a thin beam of length s. The braids of the two actuators on the sides were tied to the thin beam. The length of the actuators along their center lines l_i was related to the bending angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \alpha = 2 { \frac { s - { l_1 } } { { D_1 } } } = 2 { \frac { { l_2 } - s } { { D_2 } } } \tag { 10 } \end{align*} \end{document}

where D_i is the diameter of the actuator cross-section. We assumed that the actuator radius was approximately the distance to the actuator centerline from the centerline of the thin beam.

McKibben muscles increase in diameter D as they contract. By approximating the shape of the McKibben muscle as a cylinder, the diameter D and centerline length l of the actuator were written in terms of the braid angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta$$ \end{document} , the length b of fibers that make up the helices in the braid, and the number n of turns around the axis that each helix makes^39,43 \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{split} { D = { \frac { b \sin \theta } { \pi n } } } \hfill \\ { l = b \cos \theta . } \hfill \\ \end{split} \tag { 11 } \end{align*} \end{document}

These expressions were substituted for actuator 1 in Eq. (10) to relate the angle of the manipulator \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} to the braid angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _1}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \alpha = 2 \pi { n_1 } \left( { \frac { s } { { { b_1 } \sin { \theta _1 } } } - \frac { 1 } { { \tan { \theta _1 } } } } \right) . \tag { 12 } \end{align*} \end{document}

The inductance of the Smart Braid sensors was approximated by considering the circuit formed by the wires to be a long solenoid. ³⁹ The solenoid has a fixed number of turns N and a core with magnetic permeability \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mu$$ \end{document} . The inductance L of such a circuit will grow as the sensor contracts in length l and grows in cross-sectional area A \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} L = \mu \frac { { { N^2 } A } } { l } . \tag { 13 } \end{align*} \end{document}

The number of turns in the solenoid circuit ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$N = n{n_h}$$ \end{document} ) is the product of the number of turns n made by each helix and the number of helices n_h.

This approximation is appropriate even when the actuator is bent along its length. In this case, the circuit shape can be considered a segment of a toroidal inductor. The equation for the inductance of a toroid is identical to that of a straight long solenoid with a length equal to the circumference of the toroid centerline.

Equation (11) was used to express the inductance predicted by Eq. (13) in terms of the braid angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} L = { \frac { \mu n_h^2b } { 4 \pi } } \sin \theta \tan \theta . \tag { 14 } \end{align*} \end{document}

Inverting Eq. (12) results in the following expression for the braid angle on actuator 1 \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _1}$$ \end{document} in terms of the bend angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{split} { { \theta _1 } } { = 2 \mathop { \tan } \nolimits^ { - 1 } \left( { \gamma \left( \alpha \right) } \right) } \hfill \\ { \gamma \left( \alpha \right) } { = \displaystyle { \frac { \sqrt { { \alpha ^2 } - { { \left( { { \frac { 2 \pi { n_1 } s } { { b_1 } } } } \right) } ^2 } + { { \left( { 2 \pi { n_1 } } \right) } ^2 } } + \alpha } { \left( { 2 \pi { n_1 } } \right) \left( { \frac { s } { { { b_1 } } } + 1 } \right) } } . } \hfill \\ \end{split} \tag { 15 } \end{align*} \end{document}

Note that the following expressions are true: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{split} { \sin \left( { 2 \mathop { \tan } \nolimits^ { - 1 } \left( \gamma \right) } \right) } { = { \frac { 2 \gamma } { { \gamma ^2 } + 1 } } } \hfill \\ { \tan \left( { 2 \mathop { \tan } \nolimits^ { - 1 } \left( \gamma \right) } \right) } { = { \frac { - 2 \gamma } { { \gamma ^2 } - 1 } } . } \hfill \\ \end{split} \tag { 16 } \end{align*} \end{document}

Substituting Eq. (15) into Eq. (14) with the identities in Eq. (16) results in the following: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { L_1 } \left( \alpha \right) = \left( { { \frac { \mu n_h^2 { b_1 } } { 4 \pi } } } \right) \left( { { \frac { - 4 { \gamma ^2 } \left( \alpha \right) } { { \gamma ^4 } \left( \alpha \right) - 1 } } } \right) . \tag { 17 } \end{align*} \end{document}

Equation (17) is defined for the inductance of the sensor on actuator 1. The inductance L₂ of the sensor on actuator 2 is given by evaluating Eq. (17) with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$- \alpha$$ \end{document} .

In the range we are interested in, Eq. (17) can be approximated as a quadratic function. We numerically evaluated Eq. (17) from −90° \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\le \alpha \le$$ \end{document} 90° ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${b_1} = $$ \end{document} 31 cm, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${n_h} = 16$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${n_1} = \, 3.3$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s = $$ \end{document}  26 cm, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mu = $$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$4 \pi \times {10^7}$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm ^H/_m}$$ \end{document} ). A quadratic regression to the inductance response over this range had an R² value of 0.99999. Thus the inductance response of actuators 1 and 2 can be approximated as identical quadratic functions where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${L_1} \left( \alpha \right) = {L_2} \left( { - \alpha } \right)$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{L_1} \left( \alpha \right) }{ = f \left( \alpha \right) \approx { \lambda _0} + { \lambda _1} \alpha + { \lambda _2}{ \alpha ^2}} \hfill \\ {{L_2} \left( \alpha \right) } { = f \left( { - \alpha } \right) \approx { \lambda _0} - { \lambda _1} \alpha + { \lambda _2}{ \alpha ^2}.} \hfill \\ \end{matrix} \tag{18} \end{align*} \end{document}

In the difference of L₁ and L₂, the quadratic terms are canceled and the linear terms are doubled \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {L_1} - {L_2} = 2{ \lambda _1} \alpha . \tag{19} \end{align*} \end{document}

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} can then be expressed as a linear function of L₁ and L₂ \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \alpha \left( { { L_1 } , { L_2 } } \right) = \frac { 1 } { { 2 { \lambda _1 } } } { L_1 } - \frac { 1 } { { 2 { \lambda _1 } } } { L_2 } . \tag { 20 } \end{align*} \end{document}

In practice, the inductance response of the two Smart Braid actuators will not be identical (Fig. 12). Due to the manual fabrication process, the sensors and their connection to the flexible spine will be different. They may differ in the arc length s they enclose, the effective length b, or the number n of turns in their helices. These anticipated differences lead us to assume that the inductance response of two actuators will be best approximated by different quadratic functions: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{L_1}} { = { \lambda _{01}} + { \lambda _{11}} \alpha + { \lambda _{21}}{ \alpha ^2}} \hfill \\ {{L_2}} { = { \lambda _{02}} - { \lambda _{12}} \alpha + { \lambda _{22}}{ \alpha ^2}.} \hfill \\ \end{matrix} \tag{21} \end{align*} \end{document}

Still, the quadratic terms can be canceled \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { L_1 } - { \frac { { \lambda _ { 21 } } } { { \lambda _ { 22 } } } } { L_2 } = \left( { { \lambda _ { 01 } } - { \frac { { \lambda _ { 21 } } } { { \lambda _ { 22 } } } } { \lambda _ { 02 } } } \right) + \left( { { \lambda _ { 11 } } + { \frac { { \lambda _ { 21 } } } { { \lambda _ { 22 } } } } { \lambda _ { 12 } } } \right) \alpha . \tag { 22 } \end{align*} \end{document}

This leaves a form of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} , which is a linear equation in two variables \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \alpha _{est}} = {a_{c1}}{L_1} + {a_{c2}}{L_2} + {b_c}. \tag{23} \end{align*} \end{document}

Calibration, continuum manipulator

The control variable for the continuum manipulator was the pressure difference \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta P$$ \end{document} between the two actuators. From this difference, the desired pressure in each actuator was computed such that there always remained a minimal baseline pressure of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${P_{base}}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{P_{1 , des}}} { = \left( { \begin{matrix} {{P_{base}} + \Delta P , } & \; { \Delta P > 0} \\ {{P_{base}} , } & { \Delta P \le \, 0} \\ \end{matrix} } \right.} \hfill \\ {{P_{2 , des}}} { = \left( { \begin{matrix} {{P_{base}} , } & { \Delta P \ge \, 0} \\ {{P_{base}} - \Delta P , } & \; { \Delta P < 0.} \\ \end{matrix} } \right.} \hfill \\ \end{matrix} \tag{24} \end{align*} \end{document}

The value of baseline pressure \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${P_{base}}$$ \end{document}  = 0.05 MPa was selected to roughly correspond to the onset of actuator motion under no-load conditions.

To calibrate the continuum manipulator, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta P$$ \end{document} was increased and decreased linearly between ∼−0.1 MPa and 0.1 MPa five times over the course of 10 min. Because of the slowly changing pressure values, the dynamics of the actuator motion were neglected. The calibration pressures were assumed to correspond to steady-state measured angles. These data were processed to identify a relationship between \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta P$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} . The data were regressed to a linear approximation of the form \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \alpha = {a_{cp}} \Delta P + {b_{cp}}. \tag{25} \end{align*} \end{document}

The identified coefficients of Eq. (25) are listed in Table 4. The nonzero value of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${b_{cp}}$$ \end{document} is indicative of the asymmetry in the system. The angle to pressure relationship exhibited hysteresis and nonlinearity (Fig. 16). Different pressure values often resulted in the same tip angle. This is apparent in the large degree of variation in the angle that is not captured by the linear model ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${R^2} = 0.826$$ \end{document} ).

The calibration data were also used to identify a relationship between the continuum angle and the measured inductance of the Smart Braid actuators (Fig. 12). A two-variable linear regression was used to identify the coefficients of Eq. (23). They are listed in Table 5. The RMS of the residual error was \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${1.17^ \circ }$$ \end{document} (Fig. 13).

OPEN IN VIEWER

FIG. 12.

Shown are the inductance values of the two actuators against the corresponding tip angle of the continuum manipulator. Inductance measurements are roughly quadratic with respect to the tip angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} . Color images available online at www.liebertpub.com/soro

OPEN IN VIEWER

FIG. 13.

Shown are the estimated values of the tip angle ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \alpha _{est}} = {a_{c1}}{L_1} + {a_{c2}}{L_2} + {b_c} = 31.93{L_1} - 45.11{L_2} + 77.08$$ \end{document} ) from the inductance values used in the calibration. These values are compared against the tip angle recorded by the camera. The quadratic terms in the sensor response of the individual Smart Braids (Fig. 12) are fully compensated by a linear combination of both inductance measurements. Color images available online at www.liebertpub.com/soro

Feedback control of continuum manipulator

The Smart Braid feedback controller used inductance values and Eq. (23) to estimate the manipulator tip angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \alpha _{est}}$$ \end{document} (Fig. 14a). This estimate was compared against the reference angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \alpha _{des}}$$ \end{document} to calculate the feedback error e \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} e = { \alpha _{des}} - { \alpha _{est}}. \tag{26} \end{align*} \end{document}

This error e was scaled by the gain k_c to calculate a desired angular rate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \dot \alpha _{des}}$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \dot \alpha _{des}} = {k_c}e. \tag{27} \end{align*} \end{document}

The desired angular rate was again scaled by the inverse of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${a_{cp}}$$ \end{document} to calculate a desired rate of pressure difference change \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta {\, \dot{ P}_{des}}$$ \end{document} according to the time derivative of Eq. (25) \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \Delta { \, \dot { P } _ { des } } = { \frac { { { \dot \alpha } _ { des } } } { { a_ { cp } } } } . \tag { 28 } \end{align*} \end{document}

The desired rate of pressure difference change \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta {\, \dot { P}_{des}}$$ \end{document} was then integrated numerically in the controller to calculate the desired pressure difference \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta {P_{des}}$$ \end{document} . The corresponding actuator pressure values were calculated with Eq. (24) and sent to pressure-controlled valves.

For comparison, an open-loop, feedforward controller was implemented that used the inverse of Eq. (25) to generate pressure commands for the actuators (Fig. 14b).

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FIG. 14.

We compared the performance of two controllers on the continuum manipulator: (a) the Smart Braid feedback controller uses tip angle estimates from the inductance-based, Smart Braid sensors and (b) a feedforward controller for the actuator pressures.

Experimental implementation, continuum manipulator

The continuum manipulator used the same data acquisition and pneumatic control hardware as the revolute joint. The manipulator was fabricated by fastening the outer, conductive braids of the actuators to a flexible spine. The spine consisted of two (0.83 mm thick) strips of Delrin plastic. To establish a ground truth, the angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} of the actuator tip was additionally measured by visually tracking the motion of two points at the actuator tip. The angle of these points was computed relative to two-fixed points on the ground plane. The point positions were recorded with a high-frame-rate camera (120 fps) placed above the manipulator (with the camera's visual field parallel to the plane of actuation). The system pressure was limited to ∼0.20 MPa. As with the revolute joint, the Smart Braids were connected in parallel to 390 pF capacitors. For this system, the target inductance sampling rate was 250 Hz.

The value of the gain k_c used in the inductance feedback loop was 5 \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \rm{s}}^{ - 1}}$$ \end{document} . The performance of controllers was evaluated with the same pseudorandom step input used with the revolute joint. Each controller was evaluated thrice. The steady-state tracking was evaluated over the latter half of each step. That is, the last 5 s of a step lasting 10 s.

Results, continuum manipulator

The inductance feedback controller was able to track the reference signal with a smaller steady-state error than the pressure feedforward controller. Visually tracked tip angle trajectories from typical controller trials are shown in Figure 15. The steady-state RMS error of the inductance feedback controller had an average value of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${1.25^ \circ }$$ \end{document} and an SD of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${0.63^ \circ }$$ \end{document} . On the other hand, the pressure feedforward controller had larger and less consistent steady-state errors (mean of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${14.98^ \circ }$$ \end{document} , SD of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${9.58^ \circ }$$ \end{document} ). Figure 16 shows the pressure differences and tip angles recorded during the controller trials alongside the calibration data.

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FIG. 15.

Shown are the tip angles of the continuum manipulator as recorded by the camera during two trials. The inductance feedback controller allowed the continuum manipulator to track the reference input with an RMS of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${1.25^ \circ }$$ \end{document} in the steady-state error (evaluated in the last 5 s of the step). Supplementary Video S1 (Supplementary Data are available online at www.liebertpub.com/soro) includes a recording of the inductance feedback trial. Color images available online at www.liebertpub.com/soro