Compliance Model-Based Contact Force Control for Soft Continuum Robots

Abstract

Soft robots are increasingly being explored and developed in various settings that demand safe and adaptable interactions between robots and their environments. In addition, soft robots exhibit passive compliant behavior and generate continuous deformations when engaging with the environment. This imposes challenges on achieving active, on-demand interaction force control, especially when feedback force-sensing devices are not available. Consequently, there is a need to explore new model-based force control paradigms for soft robots. In this article, we propose a (quasi-)static force control approach for soft robots based on compliance modeling, avoiding the necessity for feedback control loops or extensive training data collection. The proposed approach can deliver contact force control along three Cartesian axes when the robot is actuated into various configurations. The compliance matrix is derived from the robot configuration, which allows the calculation of desired deflection displacements needed to generate on-demand forces. The resulting force control is achieved by solving inverse kinematics problems based on these deflection displacements. The efficacy of our proposed controller is validated through experiments with both one- and two-segment pneumatic-driven soft continuum robots. The results demonstrate effective static force control performance, with mean control errors below 5% of the desired peak forces.

Keywords

soft continuum robots pneumatic actuators force control compliance model

Introduction

Soft robots, made of compliant materials, offer inherent safety when interacting with their environment.^1,2 For traditional rigid-link robots, interaction forces in the task space are provided and regulated by controlling torque or displacement in joint space, often using impedance or admittance control strategies.³ In contrast, soft robots generate interaction forces through self-deformation. The deformability of soft robots mitigates contact uncertainties and results in more adaptive robot–environment interactions, which is especially valuable for innovations in collaborative robots⁴ and next-generation medical robots.^5,6 However, the inherent robot compliance that makes soft robots advantageous also poses challenges for actively controlling and applying precise forces in desired directions.⁷

To regulate the Cartesian forces of soft continuum robots, force control techniques traditionally used in rigid-link robots can be adopted. A prominent force control approach is discretizing soft robots into finite elements and regulating virtual joint forces from each element using Jacobian projection.⁸ For instance, an intrinsic force-sensing approach was proposed to estimate the wrench exerted at a tendon-driven continuum robot tip by monitoring the actuation forces.⁹ Consequently, this estimated wrench is further advanced as feedback information to implement closed-loop force control.¹⁰ Specifically, a hybrid motion and force control approach was proposed for tendon-driven continuum robots, achieving force regulation or contact surface estimation from sensed forces in the task space. In these cases, actuation forces are typically obtained from force transducers in tendon-driven or parallel continuum robots.^11,12 However, for fluidic-driven robots, additional considerations are required to convert fluidic pressure or volume into generalized actuation forces or torques.¹³ An inverse dynamics controller was proposed to regulate the in-plane Cartesian impedance for pneumatic-driven soft robots interacting with their environment, based on the assumption of piecewise constant curvature.¹⁴ For hydraulic-driven soft robots, a combination of fluid volume and pressure can be used for intrinsic force sensing and control in soft parallel robotic systems.¹⁵

In addition to model-based force sensing and control, technological advances have enabled the integration of force-sensing devices into soft robots. For instance, forces exerted on the robot can be measured directly using attached force/torque (F/T) sensors.¹⁶ However, since F/T sensors are typically rigid, their embedding in soft robots presents challenges. In contrast, fiber optics, which are inherently flexible,¹⁷ can measure forces by detecting strains. For example, stretchable sensors based on optical waveguides have been developed for applications such as prosthetic hands.¹⁸ In addition, advances in materials science have led to the development of flexible force sensors using technologies such as piezo-electric polymers,¹⁹ liquid metals,^20,21 and pneumatic fluids.²² Moreover, fusion of sensed information and models offers new perspectives on interaction force estimation and regulation of soft robots.^23,24 A vision-based external force-sensing approach was proposed in combination with finite element method, where the intensities and the locations of the external forces can be estimated.²⁵ Similarly, the deformation of soft robots derived from motion tracking can be utilized to achieve force and motion control,²⁶ as well as estimate and attenuate external force disturbances.^27,28

Another force-sensing and control paradigm for soft robots is the data-driven approach.^29–31 One such method is a sensorless force and displacement estimation technique that relies solely on fluidic pressure and volume in a pneumatically driven soft actuator.³² In addition, a learning-based closed-loop force control strategy was developed by integrating embedded soft sensors with recurrent neural networks, allowing the system to effectively handle significant drift and hysteresis in feedback signals.³³ It is important to note that the performance of data-driven approaches depends heavily on the quality and diversity of the training data, which can also limit their generalization and efficacy.

Achieving on-demand force control in specific directions poses significant challenges,³⁴ particularly in applications where size constraints or the inherent softness of the robot³⁵ hinder the integration of force sensors, such as in minimally invasive surgery.³⁶ In these cases, developing force control strategies that do not rely on force-sensing devices or large datasets for training could greatly expand the applicability of soft robots.

As highlighted in Table 1, this work advances the model-based (quasi-)static force control techniques for pneumatic-driven soft continuum robots. Specifically, our approach builds on a compliance model. To achieve force control, we can obtain the compliance matrix under a certain robot configuration determined by using the Cosserat rod model. Using this compliance matrix and the desired force, we calculate the target deflection, which in turn defines the new robot configuration. The actuation pressure required to achieve this configuration is then obtained by solving the inverse kinematics of the Cosserat rod model, which we implement using a shooting method. The effectiveness of the proposed method is experimentally validated using one- and two-segment soft robots, demonstrating satisfactory force control accuracy. For example, mean force errors are below 5% of the desired peak forces.

Table 1.

Comparison of Force Control Techniques for Soft Robotic Systems

References	Force-sensing device	Force-sensing principle	Force control principle	Control error^a	Multiaxis force control
Lindenroth et al.¹⁵	Required	Fluidic pressure and volume	Closed-loop PID control	≈10% to 20%	Yes
Tawk et al.²²	Required	Fluidic pressure	Closed-loop PID control	—	No
Wang et al.²⁶	Required	Vision	Closed-loop PD control	—	No
Yang et al.²⁹	Required	Optical tactile sensing	Learning-based, closed-loop control	<2.17%	No
Thuruthel et al.³³	Required	Strain sensors	Learning-based, closed-loop control	≈10%	No
Li et al.³⁷	Required	Flexible force sensor	Closed loop control	6–13%	No
Our work	Not required	Not required	Compliance model-based, open-loop control	<5%	Yes

The control error is defined as the ratio of the force error to the desired force value.

In summary, the key contributions of this work are: (1) The development and validation of a novel on-demand Cartesian force control method for soft continuum robots (see Fig. 1a) that solely relies on the robot’s compliance model, without the need for integrated feedback loops such as force estimation or sensing. (2) Various multiaxis force control strategies are realized. For instance, robots’ tip forces along the x-, y-, and z-axes can be controlled under different robot configurations. Notably, when implementing the force controller on a two-segment robot, both uniform force control and distal-only force control can be achieved (see Fig. 1b).

FIG. 1.

(a) Model-based, on-demand force control architecture based on compliance modeling and solving inverse kinematics problems. (b) Various force control modes, including force control for a one-segment robot, uniform force control, and distal-only force control for a two-segment robot.

Materials and Methods

Soft robotic prototype

The main body of the soft robot is made of a highly deformable elastomer (Ecoflex 00-50, SmoothOn) with a pneumatic-driven principle.^16,38 To seal the actuation chambers at both ends, a stiffer material, Dragon Skin 30, is used. The robot has six individually reinforced circular chambers. Since two adjacent chambers are connected as one pair, $P_{i}$ is used to denote the pressure in the $i$ th chamber pair. As illustrated in Figure 1a, $i \in [1, 3]$ and $i \in [4, 6]$ denote the pressure in the proximal (close to the fixed base) and distal segments, respectively. As such, for a one-segment robot, the pressure vector $P = [P_{1}, P_{1}, P_{2}, P_{2}, P_{3}, P_{3}]$ ; for a two-segment robot, the pressure vector $P = [P_{1}, P_{1}, P_{2}, P_{2}, P_{3}, P_{3}, P_{4}, P_{4}, P_{5}, P_{5}, P_{6}, P_{6}]$ .

Architecture of compliance model-based force control

The compliance model-based force control architecture is shown in Figure 1a. A compliance model is used to compute both the robot’s compliance and its configuration under an initial actuation pressure. Once the compliance matrix $C^{o}$ at a given robot configuration $g_{c}$ is obtained, the target deflection can be used to get the new robot configuration, named deformed configuration $g_{d}$ , using this compliance matrix and the desired force $f_{d}$ . The actuation pressure required to achieve this configuration, and thereby generate the desired force, is then determined by solving the inverse kinematics. Accordingly, the following sections elaborate on the compliance model, derivation of robot deflection, and the formulation of the force control problem as inverse kinematics problems.

Configuration-dependent compliance modeling

The compliance behavior of continuum robots varies under different robot configurations.^12,34,39 In this work, the configuration-dependent compliance of the robot is computed with its configuration, which is modeled using the Cosserat rod model. This integrated approach enables efficient computation of the compliance matrix at any arbitrary point along the robot’s body and across different configurations, without increasing the complexity of solving the ordinary differential equations (ODEs) in the Cosserat rod model. The connection between the compliance model and the Cosserat rod model lies in the adjoint matrix ${Ad}_{o b}$ in (1). Specifically, the configuration-dependent compliance can be described by a set of ODEs with respect to the curve $s$ along the robot’s backbone,⁴⁰ which yields

\{\begin{array}{l} p_{s} (s) = R (s) v (s), \\ R_{s} (s) = R (s) \hat{u} (s), \\ n_{s} (s) = - f_{e} (s) + f_{P} (s), \\ m_{s} (s) = - {\hat{p}}_{s} (s) n (s) - l_{e} (s) + l_{P} (s), \\ C (s)_{s}^{o} = {Ad}_{o b}^{- T} (s) c^{b} {Ad}_{o b}^{- 1} (s), \end{array}

(1)

where

s \in [0, L]

and

L

is the robot’s length.

\hat{(\cdot)}

is the mapping from

R^{3}

s o (3)

p_{s} (s)

R_{s} (s)

n_{s} (s)

, and

m_{s} (s)

are the derivatives of the position vector

p (s)

, the rotation matrix

R (s)

, the internal force

n (s)

, and the internal moment

m (s)

, respectively.

v (s)

and

u (s)

are the local strain and the curvature vectors.

f_{e} (s)

and

l_{e} (s)

are the distributed external force and moment.

c^{b} = diag [c_{s e}^{b}, c_{b t}^{b}]

is compliance density matrix in the body frame.

c_{s e}^{b} = diag [G A, G A, E A]^{- 1}

c_{b t}^{b} = diag [E I_{x}, E I_{y}, G J_{z}]^{- 1}

G

and

E

are the shear and Young’s moduli.

I_{x}

I_{y}

, and

J_{z}

are the second moment of area around the

x

y

,

and

z

- axes.

{Ad}_{o b}

is the adjoint matrix.⁴⁰

f_{P} (s)

and

l_{P} (s)

are the distributed force and moment from pressurization, with analytical forms reported in Supplementary Appendix SA1. Equation (1) describes both the robot configuration (

p (s)

R (s)

) and the general

6 \times 6

compliance matrix

C (s)^{o}

written in the global frame. Moreover, to accommodate a nonlinear strain–stress relation, a pressure-dependent dynamic modulus

E = h (P)

can be introduced,⁴¹ where

h (P)

is a nonlinear function [see Equation (10)].

To solve Equation (1), boundary value problems can be formulated considering force and moment balances at robot’s tip. Shooting methods can then be employed to derive solutions via numerical optimization.^40,42 To obtain the configuration-dependent compliance of soft robot, the objective function is defined as

\begin{matrix} \min_{g (0)} | | e_{n} | |^{2} + | | e_{m} | |^{2}, g (0) = [n_{0}, m_{0}] . \\ s . t . e_{n} = n (t^{-}) - F_{P} (t^{-}) - F_{e} (t^{+}), \\ e_{m} = m (t^{-}) - m_{P} (t^{-}) - m_{e} (t^{+}), \\ b_{c} = 0 (f o r a t w o - s e g m e n t r o b o t), \end{matrix}

(2)

where

g (0)

is the initial guessing values and includes the unknown force

n_{0}

and moment

m_{0}

at the manipulator’s base. The superscripts ⁻ and ⁺ denote the left and right limits.

F_{e} (t^{+})

and

m_{e} (t^{+})

are the external applied tip force and moment.

F_{P} (t^{-})

and

m_{P} (t^{-})

are the total pressurization force and moment at the manipulator’s tip in the global frame.

b_{c}

includes intermediate boundary conditions. Analytical forms of

F_{P} (t^{-})

m_{P} (t^{-})

, and

b_{c}

are summarized in Supplementary Appendix SA1. The fourth-order Runge–Kutta scheme is adopted for integrating (1).

In summary, the input to the compliance model is the actuation pressure $P$ , and the outputs are the compliance matrix $C (s)^{o}$ , the position vector $p (s)$ , and the rotation matrix $R (s)$ . The compliance matrix is used to calculate the desired deflection subject to the desired force, detailed in the following section.

Robot deflection derivation from desired force

The global tip compliance matrix can $C^{o} (L)$ be transformed to the body frame via

C^{b} (L) = {Ad}_{o b}^{T} C^{o} (L) {Ad}_{o b},

(3)

where

C^{b} (L)

is the tip compliance matrix expressed in the body frame. Considering a desired tip force

f_{d}^{o}

expressed in the global frame, the resulting wrench

W^{b}

expressed in the body frame is

W^{b} = [\begin{matrix} R^{T} & 0_{3 \times 3} \\ 0_{3 \times 3} & R^{T} \end{matrix}] [\begin{matrix} f_{d}^{o} \\ 0_{3 \times 1} \end{matrix}] .

(4)

As a result, the desired deformed configuration of the robot, that is, the homogeneous transfer matrix $g_{d}$ , can be obtained using exponential map, which yields

g_{d} = g_{c} e^{{(T^{b})}^{^}} = g_{c} e^{{(C^{b} W^{b})}^{^}},

(5)

where

T^{b}

is the resulting twist in the body frame.

{(T^{b})}^{^} \in R^{4 \times 4}

, which writes the twist in the form of an isomorphism matrix in Lie algebra.⁴⁰ The deformed tip position vector

p_{d}

can be extracted from

g_{d}

Force control formulated as inverse kinematics problems

Solving (1) to (2) produces robot configurations and the compliance matrix based on actuation pressure. Equation (5) then calculates the deformed tip position $p_{d}$ using the desired force vector $f_{d}^{o}$ . By solving the inverse kinematics problem from $p_{d}$ , the desired actuation pressure can be obtained to achieve the desired force. Various force control modes can be achieved by formulating different objective functions. For the inverse kinematics, the initial guessing values are

g (0) = [n_{0}, m_{0}, P],

(6)

where

P

is the desired pressure vector.

P \in R^{3}

and

P \in R^{6}

for one- and two-segment robots, respectively.

For multisegment soft robots, actuating different segments can produce various force control modes, which is beneficial for varying grasping performances.⁴³ As such, this work demonstrates two force control modes for a two-segment robot, including uniform force control and distal-only force control modes. As illustrated in Figure 1b, various force control modes can be achieved and are formulated as follows.

Force control for a one-segment robot

The objective function yields

\begin{matrix} \min_{g (0)} | | e_{n} | |^{2} + | | e_{m} | |^{2} + | | e_{t} | |^{2} . \\ s . t . e_{n} = n (t^{-}) - F_{P} (t^{-}) - F_{e} (t^{+}), \\ e_{m} = m (t^{-}) - m_{P} (t^{-}) - m_{e} (t^{+}), \\ e_{t} = p_{d} - p (t), t = L_{1} . \end{matrix}

(7)

The desired position $p_{d}$ is derived from the deformed robot configuration in (5).

Uniform force control for a two-segment robot by coupled pressure

For the uniform force control mode, the pressure from two robot segments is controlled by coupled pressure, that is, $P_{1} = P_{4}$ , $P_{2} = P_{5}$ and $P_{3} = P_{6}$ . The objective function is

\begin{matrix} \min_{g (0)} | | e_{n} | |^{2} + | | e_{m} | |^{2} + | | e_{t} | |^{2} . \\ s . t . e_{n} = n (L^{-}) - F_{P} (L^{-}) - F_{e} (L^{+}), \\ e_{m} = m (L^{-}) - m_{P} (L^{-}) - m_{e} (L^{+}), \\ e_{t} = p_{d} - p (t), t = L_{1} + L_{k} + L_{2}, \\ b_{c} = 0, P_{1} = P_{4}, P_{2} = P_{5}, P_{3} = P_{6} . \end{matrix}

(8)

Distal-only force control for a two-segment robot by decoupled pressure

In this force control mode, the pressure from two robot segments is controlled independently; meanwhile, the tip force is only generated by the second robot segment.

\begin{matrix} \min_{g (0)} | | e_{n} | |^{2} + | | e_{m} | |^{2} + | | e_{t} | |^{2} + | | e_{k} | |^{2} . \\ s.t. e_{n} = n (L^{-}) - F_{P} (L^{-}) - F_{e} (L^{+}), \\ e_{m} = m (L^{-}) - m_{P} (L^{-}) - m_{e} (L^{+}), \\ e_{t} = p_{d} - p (t), t = L_{1} + L_{k} + L_{2}, \\ e_{k} = Δ p (L_{1}), b_{c} = 0, \end{matrix}

(9)

where

Δ p (L_{1})

denotes the variation between the desired midpoint position and the calculated midpoint position from the forward kinematics of a two-segment robot. In the inverse control, the nonlinear least-square problems from (2) and (7)–(9) are solved using the trust-region approach. The termination tolerance of the summed squared residual errors is set as

10^{- 6}

.⁴¹

Experiment hardware

The chamber pressure of robots is regulated by proportional pressure regulators (Camozzi K8P). A compressor (HYUNDAI HY5508) pressurizes air. These pressure regulators are controlled by two Adafruit MCP4728 boards, which receive the command pressure from an Arduino Due via I2C bus. The Arduino board also monitors the chamber pressure and communicates with a host computer (Intel i7, RAM 16 GB) via a serial link. The tip force is measured by an IIT-FT17 6-DoF F/T sensor. The host computer runs MATLAB to implement the force control algorithm and processes the experimental data. Detailed descriptions of the experimental hardware are included in our previous work.⁴⁴

Results

Parameter identification

Silicone materials exhibit nonlinear, hyperelastic stress–strain behavior, influenced by various factors such as strain magnitude, temperature, and humidity.⁴⁵ This nonlinearity can be captured by an experimentally identified pressure-dependent dynamic modulus.⁴¹ To achieve this, the elongation–actuation pressure data were required. Figure 2a reports the identified parameters of robots. The elongation displacements of two segments were measured by an NDI tracker attached at the tip of the robot. Based on the averaged elongation–pressure curve (see Fig. 2b), a second-order polynomial function $h (P)$ is adopted to construct the varying modulus $E (P)$ , which yields

E (P) = β_{1} {\overset{↼}{P}}^{2} + β_{2} \overset{↼}{P} + β_{3} = h (P),

(10)

where

\overset{↼}{P}

is the averaged pressure from three chamber pairs.

β_{1}

β_{2}

, and

β_{3}

are the coefficients of the dynamic modulus. The dynamic modulus

E (P)

is reported in Figure 2c, which illustrates that the dynamic modulus monotonically decreases with pressure.

FIG. 2.

Results of the parameter identification. (a) The geometrical parameters of the robot. (b) The experimental elongation versus pressure curve. (c) The pressure-dependent modulus $E (P)$ with respect to the average chamber pressure.

Blocked force control for a one-segment robot

This experiment validates the tip blocked force control when only one chamber pair (i.e., $P_{1}$ ) is actuated. We examined eight types of time-varying desired forces, which included step force control with an increment of 0.1 N, ramp force control, and sinusoid force control with a force period $T$ of 10, 8, 6, 4, 2, and 1 s. Throughout all the tests, the maximum force magnitude was set as 1.0 N, and the duration of each test was 22 s. The tip force was measured using the F/T sensor (see Figure 3a).

FIG. 3.

Results of the tip-blocked force control for a one-segment robot: (a) snapshots for the time-varying tip force control, and the force period is 6 s. (b) Step force control. (c) Ramp force control. Sinusoid force control when the force period is (d) 10 s, (e) 8 s, (f) 6 s, (g) 4 s, (h) 2 s, and (i) 1 s.

Figure 3b–i depicts the force control results. The desired and real forces are represented in dotted black and red lines, respectively. Figure 3b illustrates that the controller can achieve satisfying step force control when the desired force increases with an increment of 0.1 N. The controlled force exhibits an overshot force of less than 0.05 N, and the settling time ranges between $0.4 and 0.5$ s. Figure 3c illustrates that the controller has a stable linearity when tracking a ramp force with a magnitude of 1.0 N. Figure 3d–i reports the results for periodical sinusoid force tracking when the force period varies from 1 to 10 s. The findings indicate a strong agreement between the desired forces and the actual forces when the force period is within the range of $4 - 10$ s. In contrast, Figure 3h–i illustrates larger deviations between the magnitudes of the desired and real forces. For instance, Figure 3i indicates that the real force magnitudes are approximately $0.1 - 0.2$ N lower and have a time delay of $0.2 - 0.3$ s, compared with the desired forces. In addition, Supplementary Video S1 reports this experiment.

Figure 4a summarizes force errors from Figure 3 using boxplots. In the case of step and ramp force tracking, the maximum force errors remain below 0.05 N, while the median errors are around 0.01 N. Compared with the maximum desired force of 1 N, the maximum and median force control errors are 5% and 1%, respectively. For the sinusoid force control, the controller demonstrates consistent accuracy when force periods are 10, 8, and 6 s, and maximum and median errors are below 0.16 and 0.05 N, respectively. As the force period decreases from 4 s to 1 s, force control errors increase. Specifically, the median errors are 0.08, 0.12, and 0.24 N for force periods of 4, 2, and 1 s, respectively. Notably, when the force period is 1 s, the maximum error reaches 0.72 N. This suggests that the controller exhibits a cutoff frequency of approximately 0.5–1 Hz.

FIG. 4.

Results of the tip-blocked force control of a one-segment robot. (a) Summary of the tip-blocked force control errors using boxplots. The color depth of the scattered points indicates the level of errors; for example, the error increases when the color varies from blue to orange. (b) Relation between the control pressure and the generated tip force using a linear regression.

Figure 4b demonstrates a strong linear relation between the control pressure and the generated tip blocked force. For example, by linear regression, the linear coefficients are 1.0, 0.92, and 0.91 N/bar for the ramp force, sinusoid force (T = 10 s), and sinusoid force (T = 4 s) tests, respectively. The shaded colors in Figure 4b represent the 95% confidence interval, indicating the level of certainty when fitting a linear relationship.

Force control along three axes for a one-segment robot

This experiment validates the tip force control along the $x$ -, $y$ -, and $z$ -axes under different robot configurations. Figure 5a illustrates the experimental setup to measure tip forces along three axes. The setup has two linear and two rotational degrees of freedom to position the force sensor, allowing the force sensor to be positioned so that the measured forces align with the sensor’s $z$ -axis. To achieve force control along different axes, Figure 5b outlines a three-stage process during the experiments. In the first stage, the robot was first actuated to one of the four configurations: (1) $P_{1} =$ 1.0 bar, (2) $P_{3}$ = 1.0 bar, (3) $P_{1} = P_{3}$ = 1.0 bar, and (4) $P_{1} = P_{2}$ = 1.0 bar. In the second stage, the robot stabilizes for a second. In the third stage, a desired constant force with a specific direction is sent to the controller, which calculates the desired command pressure. As reported in Figure 5b, the variation of the chamber pressure between the second and third stages generates the desired forces. In the experiments, the desired forces along the $x$ - and $y$ -axes are up to 0.5 N with an increment of 0.1 N. The desired forces along the $z$ -axis are up to 1 N with an increment of 0.2 N.

FIG. 5.

Results of the force control along three axes. Force measurement setup is shown in (a). The three-stage pressure sequences defined in Figure 1a for generating forces along $x$ -, $y$ -, and $z$ -axes are reported in (b), (c), and (d), respectively. In the first stage, the robot is actuated to a certain configuration. In the second stage, the actuation keeps for 1.0 s, for example, $P_{3} = 1.0$ bar in this figure. In the third stage, the force controller calculates the actuation pressure to generate a desired tip force.

Figure 6a–d reports the force control results under four configurations. The results show the forces are effectively controlled along three axes, aligning with the desired forces. In general, the real forces settle down after 2 s. The mean values and standard deviations of steady-state force errors are reported in Table 2. The results show the mean force errors along the $x$ - and $y$ -axes are less than 0.03 N, with maximum errors smaller than 0.06 N. In the $x$ - and $y$ -axes, the force controller tends to exhibit better accuracy with the increase in the desired force magnitudes. For example, the mean errors are less than 0.02 N when the desired force is 0.5 N. In contrast, the mean errors can be up to $0.04 - 0.06$ N when the desired force is 0.1 N. In addition, Table 2 shows that the force errors along the $z$ -axis are similar, with the maximum mean errors up to 0.06 N. The standard deviations of the error from Table 2 are about $0.01 - 0.02$ N in most cases. In summary, compared with the maximum desired forces, the mean errors are about 6% along the $x$ - and $y$ -axes, and these values are about 5% along the $z$ -axis. The standard deviations of errors are about $2 - 4 %$ of the maximum desired forces. The video demonstration for this experiment is reported in Supplementary Video S2.

FIG. 6.

Results of the force control along three axes under four different configurations when (a) $P_{1}$ = 1.0 bar, (b) $P_{3}$ = 1.0 bar, (c) $P_{1} = P_{3}$ = 1.0 bar, and (d) $P_{1} = P_{2} = 1.0$ bar. Forces along the $x$ -, $y$ -, and $z$ -axes are reported from the top to the bottom.

Table 2.

Results of the Force Control along Three Axes: Mean and Standard Deviation of Steady-State Force Control Errors

Configuration	Force axis	Desired forces along the x- or y-axes^a
Configuration	Force axis	0.1 [N]	0.2 [N]	0.3 [N]	0.4 [N]	0.5 [N]
$P_{1}$ = 1.0 bar	x-axis	0.0164 ± 0.0093	0.0137 ± 0.0100	0.0231 ± 0.0155	0.0188 ± 0.0119	0.0186 ± 0.0125
$P_{1}$ = 1.0 bar	$y$ -axis	0.0422 ± 0.0123	0.0364 ± 0.0111	0.0111 ± 0.0067	0.0108 ± 0.0099	0.0200 ± 0.0096
$P_{3}$ = 1.0 bar	$x$ -axis	0.0231 ± 0.0149	0.0284 ± 0.0135	0.0311 ± 0.0156	0.0112 ± 0.0112	0.0109 ± 0.0079
$P_{3}$ = 1.0 bar	$y$ -axis	0.0139 ± 0.0091	0.0165 ± 0.0106	0.0369 ± 0.0153	0.0393 ± 0.0132	0.0433 ± 0.0119
$P_{1} = P_{3}$ = 1.0 bar	$x$ -axis	0.0282 ± 0.0152	0.0248 ± 0.0158	0.0241 ± 0.0137	0.0290 ± 0.0145	0.0114 ± 0.0091
$P_{1} = P_{3}$ = 1.0 bar	$y$ -axis	0.0588 ± 0.0225	0.0437 ± 0.0198	0.0255 ± 0.0152	0.0363 ± 0.0202	0.0292 ± 0.0179
$P_{1} = P_{2}$ = 1.0 bar	$x$ -axis	0.0221 ± 0.0126	0.0398 ± 0.0184	0.0305 ± 0.0174	0.0368 ± 0.0158	0.0158 ± 0.0369
	$y$ -axis	0.0460 ± 0.0204	0.0223 ± 0.0133	0.0194 ± 0.0129	0.0209 ± 0.0132	0.0133 ± 0.0093
		Desired forces along the z-axis^a
		0.2 [N]	0.4 [N]	0.6 [N]	0.8 [N]	1.0 [N]
$P_{1}$ = 1.0 bar	$z$ -axis	0.0183 ± 0.0137	0.0242 ± 0.0121	0.0124 ± 0.0091	0.0484 ± 0.0100	0.0336 ± 0.0217
$P_{3}$ = 1.0 bar	$z$ -axis	0.0170 ± 0.0121	0.0376 ± 0.0209	0.0520 ± 0.0167	0.0127 ± 0.0111	0.0111 ± 0.0132
$P_{1} = P_{3}$ = 1.0 bar	$z$ -axis	0.0295 ± 0.0209	0.0539 ± 0.0207	0.0191 ± 0.0113	0.0611 ± 0.0176	0.0188 ± 0.0138
$P_{1} = P_{2}$ = 1.0 bar	$z$ -axis	0.0445 ± 0.0224	0.0304 ± 0.0143	0.0396 ± 0.0149	0.0261 ± 0.0183	0.0176 ± 0.0114

The reported results have the form of a ± b, where a denotes the mean error and b denotes the standard deviation of the error.

Uniform force control for a two-segment robot by coupled pressure

This experiment validates the tip force control for a two-segment robot by coupled pressure (i.e., $P_{1} = P_{4}$ , $P_{2} = P_{5}$ and $P_{3} = P_{6}$ ). In this experiment, the two-segment robot was attached to the end-effector of a Franka Emika Panda robotic arm to facilitate the force measurement (see Fig. 7). Two sets of experiments were conducted. The first experiment set focused on step, ramp, and sinusoid tip force control when the robot was not initially actuated. The second set explored the tip force control along the $x$ -, $y$ -, and $z$ -axes when the robot was initially actuated to four different configurations using different pressure sequences, that is, (1) $P_{1} = P_{4} =$ 0.4 bar, (2) $P_{1} = P_{4} =$ 0.6 bar, (3) $P_{2} = P_{3} = P_{5} = P_{6} =$ 0.4 bar, and (4) $P_{2} = P_{3} = P_{5} = P_{6} =$ 0.6 bar.

FIG. 7.

Results of the tip-blocked force control for a two-segment robot by the uniform force control when the robot is initially not actuated. (a) Step force control. (b) Ramp force control. Sinusoid force control when the force period is (c) 10 s, (d) 8 s, (e) 6 s, (f) 4 s, and (g) 2 s. (h) The summarized force control errors. (i) The linear pressure–force relation. (j) Experimental setup for the tip force measurement.

Figure 7a–i reports the tip blocked force control results when the robot is initially not actuated, and Figure 7j illustrates the experimental setup for the tip force measurement. The results demonstrate that the force controller is capable of tracking step, ramp, and sinusoid forces with a maximum force of 0.45 N, and Figure 7h reports that the median force errors for the step and ramp force tracking are below 0.015 N. For the sinusoid force tracking, the overall force errors become larger as the force period decreases from 10 to 2 s. For instance, the median errors are 0.041, 0.044, 0.060, 0.081, and 0.152 N when the time periods are 10, 8, 6, 4, and 2 s, respectively. Similar to Figure 4b, Figure 7i reports that a linear relation between generated tip force and pressure is also observed for a two-segment robot. Using a linear regression, the linear coefficients are 0.54, 0.49, and 0.47 N/bar for the ramp force, sinusoid force (T = 10 s), and sinusoid force (T = 6 s) tests, respectively. Supplementary Video S3 demonstrates this experiment set.

Figure 8 reports results of force control along three axes across four robot configurations (see Fig. 8a). Figure 8b–e and Table 3 report constant force control results, with Figure 8f–i showing the corresponding control pressures. In most cases, the average force errors along the $x$ - and $y$ -axes are between 0.01 and 0.03 N with a maximum desired force of 0.4 N. By contrast, the force errors along the $z$ -axis are between 0.01 and 0.05 N with a maximum desired force of 0.6 N. In all tests, the standard deviations of the errors are similar and between 0.01 and 0.02 N. Moreover, Figure 8f–i reports that the control pressure of two robot segments is coupled and follows a pattern of $P_{1} = P_{4}$ , $P_{2} = P_{5}$ and $P_{3} = P_{6}$ throughout all the tests. This coupling of control pressures demonstrates that both robot segments are controlled in the same manner.

FIG. 8.

Results of the tip force control along three axes for a two-segment robot by coupled pressure when the robot is initially actuated to four configurations. (a) Four robot configurations. The corresponding force control results are reported in (b)–(e). (f)–(i) report the control pressure sequences for control when controlling forces of 0.4, 0.4, and 0.6 N along the $x$ -, $y$ -, and $z$ -axes.

Table 3.

Results of the Force Control along Three Axes for a Two-Segment Robot by Coupled Pressure: Mean and Standard Deviation of Steady-State Force Control Errors

Configuration	Force axis	Desired forces along the $x$ - or $y$ -axes^a
Configuration	Force axis	0.1 [N]	0.2 [N]	0.3 [N]	0.4 [N]
$P_{1}$ = 1.0 bar	$x$ -axis	0.0132 ± 0.0082	0.0109 ± 0.0085	0.0108 ± 0.0114	0.0235 ± 0.0114
$P_{1}$ = 1.0 bar	$y$ -axis	0.0156 ± 0.0086	0.0162 ± 0.0097	0.0127 ± 0.0109	0.0305 ± 0.0091
$P_{1}$ = 0.6 bar	$x$ -axis	0.0227 ± 0.0095	0.0123 ± 0.0107	0.0377 ± 0.0152	0.0192 ± 0.0159
$P_{1}$ = 0.6 bar	$y$ -axis	0.0178 ± 0.0085	0.0141 ± 0.0091	0.0273 ± 0.0101	0.0317 ± 0.0086
$P_{2} = P_{3} = P_{5}$ = $P_{6}$ = 0.4 bar	$x$ -axis	0.0228 ± 0.0081	0.0190 ± 0.0088	0.0278 ± 0.0110	0.0237 ± 0.0134
$P_{2} = P_{3} = P_{5}$ = $P_{6}$ = 0.4 bar	$y$ -axis	0.0123 ± 0.0077	0.0624 ± 0.0087	0.0227 ± 0.0099	0.0190 ± 0.0129
$P_{2} = P_{3} = P_{5}$ = $P_{6}$ = 0.6 bar	$x$ -axis	0.0170 ± 0.0087	0.0196 ± 0.0105	0.0273 ± 0.0087	0.0381 ± 0.0146
$P_{2} = P_{3} = P_{5}$ = $P_{6}$ = 0.6 bar	$y$ -axis	0.0089 ± 0.0047	0.0106 ± 0.0092	0.0388 ± 0.0108	0.0271 ± 0.0087

		Desired forces along the $z$ -axes^a
		0.2 [N]	0.4 [N]	0.6 [N]
$P_{1}$ = 0.4 bar	$z$ -axis	0.0344 ± 0.0125	0.0271 ± 0.0153	0.0094 ± 0.0082
$P_{1}$ = 0.6 bar	$z$ -axis	0.0375 ± 0.0132	0.0487 ± 0.0172	0.0153 ± 0.0094
$P_{2} = P_{3} = P_{5} = P_{6} = 0.4$ bar	$z$ -axis	0.0315 ± 0.0090	0.0180 ± 0.0109	0.0359 ± 0.0106
$P_{2} = P_{3} = P_{5} = P_{6} = 0.6$ bar	$z$ -axis	0.0375 ± 0.0132	0.0487 ± 0.0172	0.0294 ± 0.0131

The reported results have the form of a ± b, where a denotes the mean error and b denotes the standard deviation of the error.

Distal-only force control for a two-segment robot by decoupled pressure

This experiment validates the tip force control for a two-segment robot using decoupled pressure across eight robot configurations. Here, only the pressure in the distal segment is controlled to produce desired tip forces, while maintaining chamber pressure invariant in the proximal segment. The initial actuation pressure amplitude was set as 0.5 bar. Desired forces were set as 0.25, 0.25, and 0.4 N along the $x$ -, $y$ -, and $z$ -axes, respectively.

The boxplots in Figure 9 indicate that the median force control errors along the $x$ -, $y$ -, and $z$ -axes are less than 0.03, 0.04, and 0.06 N, respectively. Figure 9 also reports the corresponding control pressures. The pressure sequences applied during 0 $-$ 2 s actuate the robot to different configurations. After 2 s, the pressure in the second robot segment ( $P_{4}$ , $P_{5}$ , and $P_{6}$ ) varies, as they are calculated by the force controller to achieve desired tip forces. In contrast, the variation of pressure in the first robot segment ( $P_{1}$ , $P_{2}$ , and $P_{3}$ , represented by dotted lines) remains relatively constant during the force generation stages. Compared with the pressure sequences in Figure 8f–i, the pressure in Figure 9 indicates that the distal-only force controller effectively decouples the pressure between the two robot segments, allowing only the distal segment to generate tip forces. The pressure amplitudes in Figure 9 reach up to 1.9 bar to generate forces of 0.25 N along the $x$ - and $y$ -axes. Supplementary Video S4 provides a demonstration of this experiment.

FIG. 9.

Results of the uniform force control along three axes for a two-segment robot by decoupled pressure. Results when (a) $P_{4} =$ 0.5 bar, (b) $P_{1} =$ 0.5 bar, (c) $P_{1} = P_{5} =$ 0.5 bar, (d) $P_{3} = P_{4} =$ 0.5 bar, (e) $P_{2} = P_{6} =$ 0.5 bar, (f) $P_{1} = P_{2} = P_{5} =$ 0.5 bar, (g) $P_{2} = P_{4} = P_{5} =$ 0.5 bar, and (h) $P_{1} = P_{2} = P_{5} = P_{6} =$ 0.5 bar. The force errors are box-plotted. Please note that two robotic segments initially bend in different planes in (c)–(g).

Discussions and Conclusion

This article proposed an on-demand force control method for soft continuum robots based on a compliance model. Our approach is capable of controlling tip forces of soft robots along the $x$ -, $y$ -, and $z$ -axes under different robot configurations. The effectiveness of the proposed force controller was validated through experiments with both one- and two-segment robots. Notably, all experiments demonstrated that ramp and step force control errors remained consistently below 0.05 N.

Tables 2 and 3 report that the standard deviation of the force control errors is about $0.008 - 0.015$ N, which is in line with the measurement resolution of the F/T sensor ( $\pm$ 0.01 N). In addition, friction between the robot tip and the force sensor may introduce noise measurement. For force control along three axes, the results demonstrate that force errors along the $z$ -axis are usually $0.02 - 0.03$ N larger than the errors from the $x$ - and $y$ -axes. This can be explained that the measured force along the $z$ -axis might not be strictly generated by the center of the robot tip. In our setup, the pressure control system achieves high accuracy, with an error of less than 0.03 bar.⁴⁴ The primary source of uncertainty in our model lies in the stiffness parameters, specifically $E I_{x}$ , $E I_{y}$ , and $E A$ . To address this, we performed experimental parameter identification to obtain the material modulus $E$ , using known design values of $L$ , $A$ , $I_{x}$ , and $I_{y}$ . Specifically, we identified a pressure-dependent modulus $E (P)$ , modeled as a second-order polynomial function, with an $R^{2}$ value of 0.984 (see Fig. 2a). As such, $E (P)$ effectively minimizes the model uncertainty arising from robot dimensions, the nonlinear material behavior, and the pressure control error. Throughout this work, the same $E (P)$ was used for all manipulators, with only one-time calibration. While this demonstrates the reusability of $E (P)$ , recalibration may become necessary due to material aging or environmental drift. Prior to experiments, diagnostic tests can be used to assess model accuracy. For instance, by applying predefined pressure to generate elongation and comparing the measured displacement with the simulated one. If the discrepancy exceeds a set threshold, parameters of $E (P)$ in Figure 2c should be updated.

Comparing Figure 7 with Figure 3, the maximum controllable tip forces are set as 1.0 and 0.45 N for one- and two-segment robots, respectively. This is attributed to the fact that the two-segment robot has higher tip compliance and lower tip force generation capability,⁴⁰ as evidenced by Figures 4b and 7i, that is, the force–pressure coefficients are about 1 and 0.5 N/bar for the one- and two-segment robots, respectively. Interestingly, a linear relationship between the tip blocked force and actuation pressure is also observed in other pneumatic-driven actuators, for example, fiber-reinforced bending actuator⁴⁶ and the PneuNets actuator.⁴⁷ The selection of initial pressure values across different experiments is constrained by a maximum allowable pressure of 2 bar, beyond which there is a risk of damaging the robot. For example, for the one-segment robot, we limited the maximum pressure to 1 bar, with peak actuation pressures remaining below 1.7 bar during the force generation. For the two-segment robot, the initial pressure is set around 0.5 bar. As shown in Figure 9d, the maximum pressure reaches approximately 1.9 bar when the initial pressure is 0.5 bar.

Figures 8 and 9 demonstrate that the second robotic segment requires higher pressure (above 1.9 bar) to generate tip forces when employing the distal-only force control, in contrast to the pressure required (less than 1.5 bar) when using the uniform force control to produce similar forces. This is due to the fact that when executing distal-only force control, only the second robotic segment contributes to the tip force generation.

In this work, we focused on the (quasi-)static force control where robot kinematics is based on the static Cosserat rod model, excluding robot dynamics. As revealed in Figures 3 and 7, the force control errors for one- and two-segment robots become larger when the frequency of desired forces increases. It is worth mentioning that soft robotic systems usually have low bandwidth (e.g., 1 Hz).⁴⁸ In our case, the bandwidth of our system is about $0.25 - 1$ Hz. Elevating the computation speed of our approach by advancing software and hardware implementation or considering the dynamics of soft robots^13,49,50 could improve the bandwidth of the force controller.

As highlighted in Table 1, our force control approach is exclusively based on the compliance model, without relying on additional force-sensing devices. In general, the mean force control errors are less than 5% of the peak desired forces. Although our approach is model-based and open-loop, it is worth mentioning that our achieved control accuracy is comparable with results from existing literature. For instance, errors are about 10% using learning-based approaches,³³ $5 - 20 %$ using force-sensing-based closed-loop control approaches.^10,15 Compared with closed-loop and learning-based force control, the deployment of feedback loop^22,26 and gathering of large training data sets^32,33 are not required in our approach. For example, our approach can be applied to soft robots in visually occluded environments, where exteroceptive measurements are needed to provide deformation information for force control.²⁶ As such, our approach is advantageous for applications where size requirements³⁶ or inherent softness of robots⁵¹ impedes sensor implementation. In addition, the identification of the dynamic modules is conveniently obtained from the elongation–pressure relation. Theoretically, the accuracy of predicted displacement derived by the compliance matrix from (5) guarantees, under the small deflection assumption⁴⁰; however, satisfying control efficacy can be achieved even when large forces are demanded (see Fig. 3).

It is noteworthy that our controller is open-loop, and, as such, cannot adaptively compensate for large environmental uncertainties. For instance, the elastic modulus of soft materials is sensitive to temperature, strain rate, and fatigue. To accommodate these uncertainties, the identified pressure-dependent modulus $E (P)$ in Figure 2c can be extended to a multidimensional table, incorporating influences of environmental parameters such as temperature and material aging. Moreover, if force sensing is available, our controller can be integrated as the inner loop within a closed-loop control framework to handle such uncertainties. Another potential limitation of our approach is its applicability to soft robots exhibiting nonuniform deformations, such as ballooning effect, which can significantly alter robots’ stiffness properties and reduce the accuracy of the compliance model.

Extending our approach to soft robots with more than two segments requires the consideration of intermediate conditions in Equations S(3) and S(4) reported in Supplementary Appendix SA1. Specifically, it is essential to ensure force and moment equilibrium, as well as kinematic continuity, at the connections between adjacent segments. Notably, increasing the number of segments introduces kinematic redundancy in robots. Consequently, additional constraints must be imposed when determining the required actuation pressures. While this added redundancy complicates control, it also enables a wider range of force control strategies. Investigating force control in soft robots with increased segments represents a promising direction for future research.

In this work, we propose a new compliance model-based force control approach intended for soft–rigid interactions, where the environment exhibits significantly higher stiffness than the soft robot. A promising direction for future research is to extend this approach to force control in interactions with compliant environments. In our approach, the interaction force is regulated by controlling desired positions, which complies with the form of stiffness control.³ As such, future work involves extending our approach to hybrid position and stiffness control^52,53 for real-world applications, such as soft medical instruments and soft grippers, where the soft manipulators might be required to delicately exert a certain amount of force on organs or objects.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This work is supported by the Springboard Award of the Academy of Medical Sciences (grant number: SBF003-1109), the Engineering and Physical Sciences Research Council (grant numbers: EP/R037795/1, EP/S014039/1, and EP/V01062X/1), the Royal Academy of Engineering (grant number: IAPP18-19\264), the UCL Dean’s Prize, UCL Mechanical Engineering, and the China Scholarship Council (CSC).

Supplemental Material

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