Cartesian Control of a 3-DOF Electro-Pneumatic Actuated Motion Platform with Exteroceptive Pose Measurement

Abstract

This paper presents a kinematic cartesian control scheme of 3 degree of freedom parallel robot driven by electro-pneumatic actuators based on exteroceptive pose measurement system. The inverse kinematics model is used to obtain the desired joint position coordinates from the time-varying trajectory given in task space. The proposal cascade control scheme in task space is based in two loops, the inner loop consisting in a decoupled joint position control and the outer loop which is designed to obtain an appropriate task space trajectory tracking. In order to avoid the on-line computation of direct kinematics an arrangement of inertial sensor and optical encoders are employed to provide the accurate pose measurement of end-effector. The experiment's results demonstrate the great performance of the proposed control scheme in industrial motion tracking application.

Keywords

Cartesian control exteroceptive measurement motion platform 3 d.o.f parallel robot

1. Introduction

The study of parallel robots has become increasingly important during last years. Hundreds of research papers have been published, prototypes have been built, new topologies invented, and consequently their applications are increasing [1].

Known as Parallel Kinematics Machines -PKM- they have excellent rigidity, fine motion, high strength-to-moving-weight ratio, high precision and repeatability; but also bring serious challenges such as limited workspace; singularities within the workspace and the kinematics and dynamic modeling [2]. From the point of view of control schemes, the control algorithms in PKM can be developed indistinctly in joint space [3] or task space [4] coordinates. In joint space control each active joint can be controlled as a decoupled independent single-input single-output control loop, with general poor compensation of the uncertainties and unmodelled dynamics. Task space control schemes have been presented for direct inverse dynamics control, with joint space dynamic model compensation [5] or task space dynamics model compensation [6]. Cartesian control schemes generally require numerical on-line computation of the end-effector pose throughout the forward kinematics, involving convergence problems and high computation time, consequently is not appropriate in a real time control [7, 8].

The applications of parallel robots in the field of motion simulators are numerous, model-based control are currently used to ensure precise path tracking, but the quality of control strongly depend on the fidelity of the model, not always achieved in the practice [9]. Nonlinear methods and intelligent algorithms are implemented in trajectory control with the disadvantages of relatively high computational effort, with limitations in industrial real-time applications and small sampling time [10].

In this paper a kinematic task space control scheme is proposed without necessity of complete dynamic model of the robot, performing the direct pose measurement of moving platform in real-time motion tracking application. The contribution showed the obtained results of trajectory control of electro-pneumatic actuated 3 degree of freedom (DOF) industrial parallel robot.

The paper is organized as follows: Introduction, following by the next part addresses the issue of principal specifications of the robot and also the description of the inverse kinematics (IK) equations. The next part is related firstly with the dynamic model of the electro-pneumatic system, and secondly with the decoupled position control scheme in joint space. Then, the cascade task space control is proposed where the position/orientation of moving platform is measured by adequate combination of inertial sensor and optical encoders. Finally, experimental results in motion tracking application are shown, demonstrating the good performance of the system.

2. Robot Description and IK model

2.1 Parallel Robot Architecture

The robotic system under study consists of 3-DOF parallel robot driven by linear pneumatic actuators. Fig. 1 shows the motion simulator and its corresponding CAD model development in MSC.Adams software subsequently used to development the simulations of different control schemes. The basic mathematical description of this system consists of the inverse kinematics expressions and the corresponding servopneumatic models, i.e. dynamic model of actuators, both used to implement the proposed kinematic control loops.

The fixed base is connected to moving platform by three pneumatic actuated kinematics chains following the RPSU-2SPS architecture. A base coordinate frame designated as OXYZ frame is fixed at the center of the base with its z-axis pointing vertically upward and the x-axis pointing backwards of the platform, according with the representation of Fig. 2. Similarly a moving coordinate frame P_X'Y'Z' is assigned to the center of the moving platform, with the z'-axis normal to the end-effector. By simplicity the directions of both z and z’ axes are pointing in the same unit vector.

Figure 1.

The 3 -DOF parallel platform and its virtual model development in Adams.

Figure 2.

The RPSU-2SPS kinematics structure of the 3-DOF motion platform.

The joints are driving by double-acting pneumatic cylinders, whose lineal displacements produce the 3 DOF of robot, consisting in two rotations around the x’ and y’ axes, represented by roll (θ) and pitch (φ) angles respectively, and linear displacement along the z’ axis (elevation) defined by the variable h. So, the moving platform can simulate different sceneries in correspondence with a virtual reality world which is shown in a LCD display located inside the cabin with capacity for 2 persons firmly attached on mobile platform.

This type of robots is developed by SIMPRO Company as industrial motion simulator for entertainment and driving simulator purposes. The most important characteristics and technical parameters of SIMPRO-PKM are summarizing in Table 1.

Table 1.

Main specifications of the parallel robot

Description	Parameters
Initial elevation of end-effector	1070 mm
Acceleration of actuators	980 mm/s²
Max. pitch and roll angles	± 18 deg
Elevation of moving platform	± 215 mm
Total mass of the robot	1034 kg
Pay-load/mass relation	2.18

Note the excellent load capacity and the relatively small workspace, both typical characteristics of parallel robots.

2.2 Inverse Kinematics Expressions

Kinematics relations of parallel robots define the relationship between the joints variables (q) and the corresponding position (x, y, z) and angular orientation (θ, φ, ψ) of center of mass of end-effector in cartesian space [11].

For n-axis parallel robot, the inverse kinematics relations Γ⁻¹ describes the joint coordinates required to reach the specified pose of mobile platform, and can be written as:

q = {[q_{1} q_{2} … q_{n}]}^{T} = Γ^{- 1} (x, y, z, θ, φ, ψ)

(1)

The forward kinematics (FK) solution Γ, could be numerically computed respect to the number of robot's joints. Then according to (1) the FK is expressing as:

x = {[x y z θ φ ψ]}^{T} = Γ {(q_{1} q_{2} … q_{n})}^{T}

(2)

Representations of actuator's displacement and closed loop vectors of active legs are shown in Fig. 3. Knowing the initial joint displacements (L_oi) and the vector L_i = ||A_iB_i|₂ which corresponds to IK solution, the joint variables can be found by evaluating q_i = ±L_i − L_oi. Then, for each kinematic chain, a vectorial function can be formulated by expressing the actuated joint coordinates as a function of cartesian coordinates (x) whose define the pose of the mobile platform.

Figure 3.

Joint displacement and closed loop vectors.

A complete inverse kinematics study of the 3-DOF parallel mechanism and validation of IK equations can be found in [12] including singularities analysis where the existence of non-singular configurations in the robot's workspace are demonstrated.

According to equation (1) is possible find the relation A_iB_i = Γ⁻¹(x) in order to derive the inverse kinematics model of the robot.

\vec{A_{i} B_{i}} |_{R_{A}} = \vec{O P} |_{R_{A}} + {}^{A}R_{B} \vec{P B_{i}} |_{R_{B}} - \vec{O A_{i}} |_{R_{A}}; i = 1…3

(3)

From vectorial formulation (3) and considering the rotation matrix ^AR_B using Roll-Pitch-Yaw convention, the following displacement-based equations can be found:

q_{1} = \pm \sqrt{{(2076 - λ_{0} - 940 c (φ))}^{2} + {(740 + h + 940 s (φ))}^{2}} - L_{o 1}

(4)

q_{2} = \pm \sqrt{{(1397 - λ_{0} + 720 c (φ) + λ_{1})}^{2} + λ_{2}^{2} + {(λ_{3} - λ_{4})}^{2}} - L_{o 2}

(5)

q_{3} = \pm \sqrt{{(1397 - λ_{0} + 720 c (φ) - λ_{1})}^{2} + λ_{2}^{2} + {(- λ_{3} - λ_{4})}^{2}} - L_{o 3}

(6)

cos(.) = c(.); sin(.) = s(.)

where:

$λ_{0} = \sqrt{1672^{2} - 1720 h - h^{2}}$

$λ_{1} = 500 s (θ) s (φ)$ ; $λ_{2} = 500 c (θ) - 500$

$λ_{3} = 500 s (θ) c (φ)$ ; $λ_{4} = 720 s (θ) + 945 + h$

Control commands are executed in the joint space while robotic motions are specified in the task space, that is why is strongly necessary to perform the inverse kinematics model -IKM- in the control scheme in order to find the corresponding sets of joints displacements given the desired position and orientation of the end-effector.

3. Joint Space Position Control

The position control of pneumatic cylinders through proportional valves focus on the kinematics of the mechanical system and on the dynamics of independent electro-pneumatic driven masses satisfies the positioning accuracy of many industrial applications [11].

In this section the design of the decoupled position joint controllers is focused on maintain the satisfactory level of robustness against dynamics interactions, as well as minimizing the disturbance effects due to possible payload variations.

Given a desired path trajectory, the required actuator displacements are calculated by the IKM, and each independent feedback loop can be designed by pole-placement method given the transfers functions of an electro-pneumatic actuators.

3.1 Dynamic Identification of Actuators

The linearized version of electro-pneumatic system is development by experimental identification. The position's transfer function Y(s) from the valve input voltage U(s) is obtained based on previous works development firstly in 2-DOF pneumatic platform [13], and subsequently extending to 3-DOF motion simulator. The model of servopneumatic actuators takes the form:

\frac{Y (s)}{U (s)} = \frac{\frac{A_{1} G_{1} / C_{1}}{τ_{1} s + 1} + \frac{A_{2} G_{2} / C_{2}}{τ_{2} s + 1}}{s (M s + b + \frac{A_{1} K_{1 y} / C_{1}}{τ_{1} s + 1} + \frac{A_{2} K_{2 y} / C_{2}}{τ_{2} s + 1})}

(7)

The time constants are computed as: τ₁ = K_1p/C₁; τ₂=K_2p/C₂.

P₁; P₂: Pressures in the cylinder chambers (Pa).

A₁; A₂: Piston area of each chamber (m²).

M: Load mass and piston (kg)

b: Viscous constant friction (N s/m)

The constants G₁, G₂ in (kg/sm2) and C₁, C₂ in (kg/sPa) are obtained by partial derivatives of the air mass flow across the flow valve: q_m = f(x; P_input; P_out). On the other hand, the constants K_1p, K_2p in (kg/Pa) and K_1y, K_2y in (kg/m) are obtained by differentiation of the state equation of the ideal gases respect to pressure and volume variables according the position of the actuator [14].

For dynamic experiments each pneumatic actuator was tested separately and PRBS input signal was applying to produce persistently excitation of servopneumatic actuators along of their complete elongation; additionally the influence of the underlap characteristic of the flow valve was considered to development the model [14]. After linearization and model reduction the dynamic behavior of electro-pneumatic system can be described by the following third order equation:

G_{D} (s) = \frac{k}{s (s^{2} + 2 ξ ω_{n} + ω_{n}^{2})}

(8)

Where ω_n and ξ are respectively the undamped natural frequency and damping ratio of the system and k is the system gain. The corresponding transfer functions obtained by close loop experimental identification are shown in Table 2.

3.2 Decoupled Position Control

Because the interaction between actuators can not be neglected without deterioration of system's performance, the design of position controllers was made considering enough robustness to minimize the effects of dynamic interaction of electro-pneumatic driven around the operation point and also the loop disturbances produced by payload variations.

The design of position controller was performed via pole placement, where the closed loop performance is dominated by complex conjugate poles with ξ = 0.7 and ω_n = 10 rad/s, and maximizing the phase margin to assure the necessary level of robustness. The transfer functions of designed controllers take the form of equation (9) and they are summarize in the Table 2.

G_{C} (s) = \frac{k_{p} (s^{2} + a_{1} s + a_{0}) (s + k_{i})}{s {(s + ω_{a})}^{2}}

(9)

Knowing the transfer functions of controllers G_C(s) and actuators G_D(S) the kinematics joint space control scheme of Fig. 4 is implemented. A low-pass second order filter F(s) with cut-off frequency of 80 Hz is inserted in the forward path in order to limit the excessive amplitude of command signals over the flow valves bandwidth.

Table 2.

Transfer functions of controllers and electro-pneumatic systems.

Active Legs	Position Controllers (Gc)	Actuator Dynamics (GD)
Actuator 1	$\frac{265 (s^{2} + 8 s + 253) (s + 3)}{s (s^{2} + 147 s + 6267)}$	$\frac{246}{s (s^{2} + 7.7 s + 253)}$
Actuator 2 and 3	$\frac{32 (s^{2} + 8 s + 1349) (s + 3)}{s (s^{2} + 147 s + 6267)}$	$\frac{2008}{s (s^{2} + 7.3 s + 1349)}$

The kinematic control scheme of Fig. 4 offers many advantages. For example, by using independent joint control, communication among different joints is saved. Moreover, since the computational load of controllers may be reduced, only low-cost hardware is required to implement industrial application in real time context and with high computation speed. Finally, independent joint control has the feature of scalability, since the joint's controllers have in general very similar formulation.

Figure 4.

Joint space position control scheme.

Each leg is actuated by pneumatic cylinder FESTO DNC-125-500 governed by a MPYE-5-3/8 proportional flow valve, joint's displacements are measured by MLO-POT-500 linear potentiometers with ± 0.01 mm of accuracy. The control algorithms are implemented in an embedded controller based on dsPIC 30F4013 dedicated microprocessor running at the 120 MHz speed, according with the hardware architecture shown in Fig. 5.

Figure 5.

Hardware architecture based on embedded controller.

Fig. 6 shows the joint's displacements during pre-filtering pulse train input signal with constant amplitude and frequency, where zero steady state error are achieved. Experiments also probed the expected robustness of the closed-loop system in the presence of dynamic interactions between actuators.

Due to unavoidable manufacturing tolerances of the mechanism, modeling errors, backlash and clearances of joints, relatively large tracking error are obtained from only implementation of decoupled position control scheme. Fig. 7 illustrated the cartesian large errors in positioning and trajectory tracking of end-effector occurred with pulse train and sinusoidal command signals respectively. Consequently the moving platform pose can not be efficiently controlled only by the decoupled joint control.

For that reason a cascade control scheme is then implemented for motion tracking purpose, by taking the advantages of two loop configuration control architecture in task space configuration.

Figure 6.

Zero steady state joint's errors during experiments with decoupled position control.

Figure 7.

Cartesian errors, in tracking (a) and positioning (b) with decoupled position control.

4. Cartesian Space Control

The objective of the task space control is designing a feedback controller that allows the execution of an end-effector motion x(t) ∊ Rⁿ that tracks the desired mobile platform motion xd(t) as closely as possible, and consequently the regulated error is the error between the measured and desired end-effector pose. So, cartesian space control ensures a direct task control and thus can be more accurate than a joint space one [8], [15].

Control approaches designed in task-space coordinates and which do not require the full dynamic model of the robot neither on-line computation of FK is proposed in this section. Consequently an appropriate exteroceptive sensorial system to provide the pose information of moving platform is also implemented.

4.1 Kinematic Task Space Control Scheme

Fig. 8 shows a schematic block diagram of the proposed task space control scheme, where the inner loop acts on position and the outer loop acts on trajectory. The inner loop was design with sufficient robustness against model's uncertainties and disturbances -see section 3-, whereas outer loop compensate the cartesian disturbances minimizing the tracking errors.

Figure 8.

Task space control scheme based in two loops.

The tracking control algorithm performed in the outer loop achieves asymptotic tracking of cartesian trajectories; however, one has complete freedom to modify the outer loop control to achieve other goals without modify the inner loop. For example, additional compensation terms may be included to enhance the robustness against to parametric uncertainties, unmodeled dynamics and external disturbances. The outer loop control may also be modified to achieve other goals, such as zero tracking error of task space trajectories, regulating motion and force, etc.

4.2 End-effector Pose Measurement

On line computing of direct kinematics of PKM in real time applications demand high performance of the computer hardware, additionally task space control schemes based on forward kinematics are affected by the numerical estimation errors and the geometrical errors, both typical characteristics of direct kinematics problem in parallel robots applications [1]. Many of these schemes have been proved by simulation or in laboratory testbed, but not commonly on industrial motion platform [3].

Based on the general idea of J. Gao [16], the combination of exteroceptive sensorial system consisting of optical encoders and inertial measurement unit is proposed to measured the pose of moving platform.

The MTi Xsens inertial motion tracking sensor is used to provide the orientation of end-effector (Fig. 9), whereas three incremental optical encoders provided the information of elevation of mobile platform, which is determined by trigonometric relations. The arrangements of sensors are shown in Fig. 12, where special care is taking in place the sensors into end-effector in order to avoid possible risks of interference of the motion.

Figure 9.

Block diagram of inertial sensor.

Fig. 10 shows the IMU and encoders locations, where both are alignment along x-axis, note that the attachment point of IMU coincides with the center of mobile platform coordinate frame P_X'Y'Z'.

Thanks to the combination of the optical encoders and inertial measurement units, a fast and accurate end-effector pose measure is available in real time for control purposes.

Figure 10.

Arrangement of sensors located on the moving platform to measure the end-effector pose.

5. Digital Control Problem Formulation

The position control under stable operation fulfilled the condition: q(t) − q_d(t) = 0; ∀ t > 0, under this assumption the digital design closed-loop control system is perform by considering a simple but satisfactory approximation of inner loop, i.e. the dynamic behavior of the decoupled position control loop is represented by one delay unit of the external loop [17], establishing that q(k)-q_d(k-1) = 0; ∀ k > 0. Consequently the control scheme of Fig. 8 is simplified by the equivalent digital system of the Fig. 11.

Figure 11.

Digital cartesian control scheme with dynamic approximation of internal loop.

Because the tracking error should be zero at steady state, a pure integrator is implemented as simple digital controller -see equation (10)-. The corresponding control law is expressed in (11).

The gain factor A₀ permits to reduce the effects of great steps in tracking control.

G_{T} (z) = A_{0} \frac{K_{I}}{(z - 1)}

(10)

Δ = K_{I} \int \tilde{ς}

(11)

Under this consideration the diagonal integral matrix is:

K_{I} = [\begin{matrix} K_{I_{1}} & 0 & 0 \\ 0 & K_{I_{2}} & 0 \\ 0 & 0 & K_{I_{3}} \end{matrix}]

(12)

Similarity as the visual control work [18], Δ can be interpreted as the coordinates increments in the task space as a result of the direct measurement of position/orientation of moving platform. Solving the inverse kinematics problem Γ⁻¹, it is possible to obtain q_d.

Task space coordinates are measured by the arrangement of linear sensors according to the simplified control scheme shown in Fig. 12, so the sensors gain matrix KM can be represented by:

K_{M} = [\begin{matrix} K_{h} & 0 & 0 \\ 0 & K_{θ} & 0 \\ 0 & 0 & K_{ϕ} \end{matrix}]

(13)

Figure 12.

Simplified control scheme.

The closed loop transfer function can be written as:

\frac{0.06 K_{I} K_{M}}{(z - 1) z} [ς_{d} (z) - ς (z)] = Y (z)

(14)

Taking the inverse Z transform and solving the state space representation is given by:

[\begin{matrix} y_{(k + 1)} \\ y_{(k + 2)} \end{matrix}] = [\begin{matrix} 0_{3 \times 3} & I_{3 \times 3} \\ - 0.06 K_{I} K_{M} & I_{3 \times 3} \end{matrix}] [\begin{matrix} y_{(k)} \\ y_{(k + 1)} \end{matrix}] = [\begin{matrix} 0_{3 \times 3} \\ 0.06 K_{I} K_{M} \end{matrix}] y_{d} (k)

(15)

Figure 13.

Root-locus plot of height control loop for I controller with closed poles for gain K_Ii K_h = 3 and sampling time of 0.06 seconds.

According equation (14) the root loci of Height control system is presented in Fig. 13, in this case the closed loop poles are selected for the gain K_hK_Ii = 3. An overdamped transient response to the step input is expected for this design. The system is stable with K_hK_Ii < 20. Similar analysis can be done for Roll and Pitch task space coordinates.

6. Experimental Results

The control scheme proposed was implemented on the 3-DOF parallel robot used by SIMPRO as motion simulator. The control algorithm of the inner and the external loops have been implemented using MATTAB/Simulink with the Real Time Workshop Toolbox and Real Time Windows Target. The end-effector elevation is measured only with encoders, whereas pitch and roll angles can be obtained via FMU or encoders. Some experiments have been developed in real motion platform, initially the system receive a pulse step variation in h_d, θ_d, and φ_d, the task space output is evaluated in proposed cartesian control scheme with joint control in the internal loop and kinematic task space control in the external loop.

The better performance of the kinematic task space control is evident, the steady state error disappear. The transient response is similar as the simulation process with Matlab/Simulink-ADAMS.

The motion tracking outputs (roll and pitch angles of mobile platform) of kinematic task space control scheme are representing in Figure 14 and 15 by using the arrangements of encoders and inertial sensor respectively. Note that the measured signals from inertial measurement unit are affected by high frequency noise. This typical characteristic of inertial sensors can be eliminated with a Butterworth second order filter designed with cut-off frequency of 10 rad/sec.

Figures 14 and 15 demonstrate the good performance in motion tracking of proposed kinematic task space control scheme, where sinusoidal reference signals are given by (15) and (16) for roll and pitch angles respectively with frequency ω = 1 rad/sec.

θ_{r e f} (t) = \sin (0.3 ω t) - \sin (ω t) - 10 \sin (1.2 ω t)

(16)

φ_{r e f} (t) = \sin (0.3 ω t) - \sin (ω t) + 4 \sin (1.2 ω t)

(17)

Figure 14.

Roll and pith orientation angles of moving platform measured with the arrangements of encoders.

Figure 15.

Filtered roll and pith angles of moving platform measured with inertial sensor.

The influence of the robot's structural dynamic effects on the trajectory errors is decreased due to the introduction of the dual control scheme. As a result, the motion precision of the moving platform is greatly improved, demonstrating the effectiveness of the implemented control system.

7. Conclusion

A motion tracking application is presented as a case of study to demonstrate the effectiveness of the cascaded control scheme applied to 3-DOF electro-pneumatic actuated motion platform. The controlled robotic system can perform trajectory tracking with enough precision according with the application, where experimental results are given to support the theory.

The design of inner digital controller by zero-pole locations satisfies the joint positioning accuracy, whereas the outer task space controller employ a feedback loop that directly minimizes cartesian errors, as result the cascade architecture providing a better control under unknown uncertainties, dynamic interactions and modeling errors, as result, good tracking accuracy for the motion system is substantially improving.

The two loops architecture gives flexibility to simulate and implement many robot's control strategies by modifying the outer loop control, while leaving the inner loop unchanged. On the other hand, the measuring system provide achievable dynamic positional and orientation accuracy in the range of control frequencies, taking as feasible end-effector pose measurement solution for the SIMPRO 3-DOF pneumatic parallel robot.

The control algorithms are relatively simple and consequently feasible to implement in real-time industrial application, where exact tracking control and excellent stability are both achieved. Future researches will be addressed to improve more robust control strategies against high pay-load variations and noise measurements.

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