Task-space model-based control of pneumatic soft actuators
Authors: Nithin S. Kumar, Joshua Gaston, D. Caleb Rucker, Eric J. Barth
Organizations: Department of Mechanical Engineering, Vanderbilt University, Nashville, Tennessee 37235 · Department of Mechanical, Aerospace, and Biomedical Engineering, The University of Tennessee, Knoxville, TN 37996
Soft actuators enable dexterous and compliant interaction, but closed-loop task-space control remains challenging due to strong nonlinearities, distributed deformation, and uncertainty in their dynamics. This paper presents a real-time dynamic-model-based task-space feedback and estimation framework based on a non-minimal coordinate discrete elastic rod model formulated in absolute coordinates with holonomic constraints. The resulting structure preserves distributed mechanics while maintaining computational efficiency through sparse system matrices, enabling real-time control with up to 10 discretized rods. A quasi-static feedforward inverse model is combined with a task-space PI controller and a dynamic observer that fuses measurement residuals as virtual forces, enabling full-state estimation from sparse sensing. The approach is experimentally validated on three planar pneumatic soft actuators with varying geometries. Across five tasks, including drawing the digits 0-9 across the workspace (3-18 mm/s tip speed), tracking periodic motion (up to 37 cm/s), cross-platform generalization, reduced sensing conditions, and real-time user-defined references, our method achieves 1.5-2.3 mm root mean square error (RMSE) for precision motions and 5.5-12.4 mm RMSE at 1-2 Hz. Results demonstrate that structured, non-minimal dynamic models can enable real-time, high-precision, moderate-bandwidth task-space control of planar soft pneumatic actuators in free space.
Figures & tables
Fig. 1: (a) The three planar soft pneumatic actuators used in this study consist of four chambers each. Tasks 1, 2, 4, and 5 are conducted with P1; Task 3 evaluates generalization using P2 and P3. P1 and P3 have the same constant cross-section and differ only in length, while the wall thickness of P2 decreases from base to tip as shown in a CAD cross-sectional view (b). Two chambers of the same side of each actuator are pressurized to 35 psi (241 kPa) resulting in bending for (c) P1, (d) P2, and (e) P3.
Fig. 2: The planar discretized soft actuator model is overlaid with the physical actuator (P2). The actuator backbone is represented by N rigid rods with rotational spring–damper elements generating elastic ( τk ) and damping ( τb ) torques. Adjacent rods are connected through holonomic constraint forces ( Fx,hc,Fy,hc ). Pneumatic actuation is modeled as generalized torques ( τext ) applied at selected rod indices.
Parameter
P1
P2
P3
Mass [g]
125.0
144.5
92.2
Length [mm]
253
253
193
ki [mNm/rad]
75.1
24.8-66.4
99.8
bi [mNm/rad/s]
0.167
0.205-0.767
0.132
TABLE I: Model Parameters
Fig. 3: The system block diagram is shown here. The reference trajectory is processed by a quasi-static inverse (feedforward) module to generate uFF . A dynamic observer integrates the constrained rod model and incorporates measurement residuals as virtual generalized forces to estimate the system state. The estimated state is used within the task-space PI control block to compute a feedback correction uFB via the local sensitivity mapping. The total input u=uFF+uFB is applied through proportional valves with low-level pressure regulation.
Fig. 4: The temporal (a) and task-space (b) response of the soft actuator to four step inputs that draw a diamond in task-space. In (a), four controllers are used. The response of the feedforward (FF) and FF+PI are then shown in task-space (b). Black Xs represent each commanded step starting at the top and moving clockwise.
Controller
RMSE [mm]
SSE [mm]
ISE [mm 2 s]
FF
30.5
15.7
3309
FF+P
23.8
11.3
2007
FF+I
19.1
1.0
1131
FF+PI
14.4
1.0
625
TABLE II: Step response average error characterization
Fig. 5: Task 1: The actuator tip is commanded to draw each of the 10 digits in 10 sec (a). The corresponding RMSE values are shown in the top left for each digit. The x and y errors ( ex,ey ) for all 10 plots are shown (b) and are always within 5 mm.
Fig. 6: Task 1: The actuator tip draws the letters “VU” spanning 25 mm in height and 55 mm in width in 25 sec. This task was accomplished with errors less than 4 mm with an RMSE of 1.9 mm (b).
Fig. 7: Task 2: The actuator is commanded to follow a 60 mm 1 Hz circle. The actual tip position, observer tip position, and reference for x and y are shown (a). The task-space trajectory for the last 4 cycles is shown (b). The observer tip closely tracks the actual tip position - a zoomed view of the y-position from t = 8.1-8.4 s is shown in (c).
Fig. 8: Task 2: Four snapshots of the actuator during one cycle of the 1 Hz circle-following task are shown (spaced 0.3 sec apart). Top row images are extracted from a phone camera and bottom row images depict the configuration of the observer during each instance.
Fig. 9: Task 2: The actuator is commanded to follow a 60 mm 1 Hz vertical line. The actual tip position, observer tip position, and reference for x and y are shown (a). The task-space trajectory for the last 4 cycles is shown (b).
Task
Task
Characteristic
Characteristic
RMSE [mm]
No.
Length [mm]
Time
1
Digits
30
10 s
1.5-2.3
1
VU
50
25 s
1.9
2
Vertical line
60
0.25 Hz
7.4
1 Hz
8.8
2
Horizontal line
60
0.25 Hz
9.2
TABLE III: Summary of Tasks 1 and 2.
Fig. 10: Task 3: The variable-curvature Pneunet (P2) is commanded to repeat Tasks 1 and 2. In (a), “VU” is drawn in 25 s with an RMSE of 3.7 mm. The corresponding error plot is shown in (b). In (c), a 60 mm, 1 Hz circle is tracked with an RMSE of 9.1 mm.
Fig. 11: Task 3: The shorter Pneunet (P3) repeats Tasks 1 and 2. In (a), “VU” is drawn in 25 s with an RMSE of 2.6 mm. The corresponding error plot is shown in (b). In (c), a 60 mm, 1 Hz circle is tracked with an RMSE of 5.5 mm. Due to the reduced workspace, trajectories are vertically shifted.
Fig. 12: Task 3: The short Pneunet (P3) is able to track a 60 mm 2 Hz circle. The x and y tip positions along with the respective phase lags are shown (a). The task-space trajectory for the last 4 cycles and RMSE for the entire 6 sec run are shown (b).
Fig. 13: Task 4: Performance under four sensing conditions — open-loop (FF), rod 6 only, rod 10 only (tip sensing), and rods 6 and 10 combined. The three reference trajectories are (a) sweep, (b) 60 mm, 0.25 Hz circle, and (c) 60 mm, 0.25 Hz vertical line. Each condition is repeated five times, and the distribution of normalized tip RMSE is shown in (d). Average absolute RMSE values for each trajectory and sensing configuration are also reported.
Fig. 14: Task 5: Real-time tracking of an arbitrary user-defined reference generated by an optical marker attached to a handheld rod is demonstrated. Representative snapshots during the 30 s trial are shown in (a). The corresponding projected reference and actuator tip trajectories in x and y are shown in (b).
Ref. (Year)
Model
Controller(s)
Platform
Size
Free-space tracking results
[ 24 ] (2017)
3-link PCC
FBL
Festo Bionic Handling Assistant
87 cm
< 3 mm static and ≈ 50 mm dynamic error (0.5 Hz)
[ 16 ] (2018)
NN
MPC
1-DOF pneumatic joint
N/A
2 ° error to 40 ° step angle tracking with 1–2 s rise time
[ 12 ] (2018)
5-link PCC
Curvature PID
Planar soft pneumatic actuator
32 cm
5 °L2 norm error (0.33 Hz)
[ 25 ] (2019)
Learned NARX model
Model-based RL policy
2-segment pneumatic soft actuator
30 cm
26 mm Cartesian error (dynamic reaching)
[ 13 ] (2022)
Lumped 2nd-order
FBL, PP/LQR, UKF
1-DOF Pneunet
10 cm
6–10 ° avg. error to angle staircase input (0.25-0.5 Hz)
[ 17 ] (2023)
Koopman
LQR
2 PMAs
37 cm, 53 cm
5–10 mm error (from reported plots) (1.1 Hz, 1.52 m/s)
TABLE IV: Recent literature summary of dynamic control of soft continuum manipulators. FBL=Feedback Linearization, PP=Pole Placement, NARX = Nonlinear Autoregressive with Exogenous Inputs, UKF=Unscented Kalman Filter, AP=Adaptive Passivity control, ANCF=Abs. Nodal Coordinate Formulation
Nithin S. Kumar (Student Member, IEEE) received the B.S. degree (Vanderbilt University), the S.M. degree (MIT), and is currently a Ph.D. candidate in Mechanical Engineering at Vanderbilt University. His research interests include the design, modeling, and control of soft/continuum robots.
Table 19
Joshua Gaston (Student Member, IEEE) received the B.S. degree (Tennessee Tech University) and the M.S. degree (UTK), and is currently a Ph.D. candidate in Mechanical Engineering at UTK. His research interests include the design and control of soft/continuum robots.
Table 20
D. Caleb Rucker (Senior Member, IEEE) is a B. Ray Thompson Professor in Mechanical Engineering at the University of Tennessee, Knoxville, where he directs the REACH lab. His research interests include design, modeling, and control of medical robots and soft/continuum robots.
Table 21
Eric J. Barth (ASME Fellow, IEEE Member) is a Professor in Mechanical Engineering at Vanderbilt University. His research interests include the design, modeling, and control of fluid power systems, soft and medical robotics.
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