A Three-Stage Offline SDRE-Based Control Framework for Human Motion Reproduction on a Suspended Bipedal Robot

Ping-Kong Huang; Chien-Wu Lan; Chin-Tien Wu; Ching-Kai Lin

cs.ROJun 5, 2025

A Three-Stage Offline SDRE-Based Control Framework for Human Motion Reproduction on a Suspended Bipedal Robot

Authors: Ping-Kong Huang, Chien-Wu Lan, Chin-Tien Wu, Ching-Kai Lin

Organizations: Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan · Department of Electrical Engineering, National Central University, Taoyuan City 320317, Taiwan

Abstract

This paper presents a three-stage offline command generation framework for reproducing human lower-limb motion on a suspended bipedal robot while matching torque trajectories computed from the robot dynamic model. First, State-Dependent Riccati Equation (SDRE) control derives the reference torque trajectory for the measured motion. Second, parameterized optimization converts this trajectory into trapezoidal joint velocity commands under motor speed and acceleration limits. Third, a proportional-integral-derivative linear quadratic regulator (PID-LQR) compensation scheme refines these commands using experimental tracking data. The platform executes the resulting profiles to reproduce human walking and squatting motions recorded by a Vicon system, allowing evaluation of tracking accuracy and repeatability. Results show that the average root mean square error (RMSE) and standard deviation (STD) of joint angles across repeated trials remain below 7° and 0.33°, respectively. Joint angle and torque trajectory comparisons show lower maximum RMSE and STD values than those for MPC and IPSO-PID in every reported case. The framework enables accurate and repeatable motion reproduction within actuator limits, providing controlled and measurable conditions that can reduce reliance on human participation and associated risks during preliminary evaluation of devices for assistive walking, gait training, and rehabilitation.

Figures & tables

Symbol	Description
$P_{1}$	Hip joint
$P_{2}$	Knee joint
$P_{c1}$	Center of mass of thigh
$P_{c2}$	Center of mass of lower leg
$l_{1}$	Thigh length
$l_{2}$	Lower leg length

TABLE I: Description of the Parameters in the Dynamic Model of the Left Lower Limb

Panel of Figure 2 — Fig. 1: (a) Overall look of the bipedal robot. (b) Dynamic model of the bipedal robot, where the symbols are defined in Table I .

Fig. 2: Flowchart of the proposed three-stage control procedure.

Panel of Figure 4 — Fig. 3: Piecewise linear velocity parameterization model

Panel of Figure 5 — Fig. 4: (a) Skeleton of lower limbs obtained from Vicon motion-capture data. (b) Simplified skeleton extracted from the Vicon motion-capture data.

Panel of Figure 6 — Fig. 5: Snapshots of the suspended bipedal robot during experiments: (a) squatting; (b) walking.

Panel of Figure 7 — Fig. 6: Comparison of reference and experimental joint angular trajectories for the suspended bipedal robot: (a) squatting; (b) walking.

Panel of Figure 8 — Fig. 7: Experimental comparison of angular trajectories under the proposed method, MPC, and IPSO-PID for (a) squatting and (b) walking.

Panel of Figure 9 — Fig. 8: Experimental comparison of model-based joint torque trajectories under the reference motion, proposed method, MPC, and IPSO-PID for (a) squatting and (b) walking.

Motion	Side	Part	$\overline{\mathrm{\mathbf{RMSE}}}$ ( ∘ )
Squatting	Left	Hip	4.2322
	Left	Knee	6.4162
	Right	Hip	4.2520
	Right	Knee	5.4709
Walking	Left	Hip	2.3029
	Left	Knee	5.5297

TABLE II: Average joint angle RMSE between the desired and experimentally reproduced trajectories.

Motion	Side	Part	Max RMSE ( ∘ )	STD ( ∘ )
Squatting	Left	Hip	4.9573	0.3220
	Left	Knee	6.8950	0.2552
	Right	Hip	4.4141	0.1371
	Right	Knee	5.6846	0.1765
Walking	Left	Hip	2.6144	0.1113
	Left	Knee	5.7767	0.1342

TABLE III: Maximum RMSE and Standard Deviation Over 10 Trials with the Same Command.

Motion	Side	Part	Max RMSE ( ∘ )			STD ( ∘ )
Motion	Side	Part	Proposed	MPC	IPSO-PID	Proposed	MPC	IPSO-PID
Squatting	Left	Hip	4.9573	23.1072	21.7738	0.3220	0.9573	1.4650
	Left	Knee	6.8950	33.3585	30.9332	0.2552	1.3899	1.9498
	Right	Hip	4.4141	22.3842	14.4919	0.1371	0.9863	0.8421
	Right	Knee	5.6846	29.2677	25.1407	0.1765	1.5468	1.8900
Walking	Left	Hip	2.6144	7.7421	8.9949	0.1113	0.2535	1.0652

TABLE IV: Comparison of Joint Angle Maximum RMSE and Standard Deviation for the Control Methods

Motion	Side	Part	Max RMSE (Nm)			STD (Nm)
Motion	Side	Part	Proposed	MPC	IPSO-PID	Proposed	MPC	IPSO-PID
Squatting	Left	Hip	0.9607	4.4610	4.3505	0.0589	0.1851	0.2856
	Left	Knee	0.4242	1.8206	1.8207	0.0187	0.0729	0.1235
	Right	Hip	0.8725	4.3836	3.0073	0.0267	0.1947	0.1782
	Right	Knee	0.3464	1.6140	1.4690	0.0149	0.0859	0.1083
Walking	Left	Hip	0.5728	1.6791	2.0110	0.0212	0.0598	0.2650

TABLE V: Comparison of Joint Torque Maximum RMSE and Standard Deviation for the Control Methods

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Abstract

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