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
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
P1
Hip joint
P2
Knee joint
Pc1
Center of mass of thigh
Pc2
Center of mass of lower leg
l1
Thigh length
l2
Lower leg length
TABLE I: Description of the Parameters in the Dynamic Model of the Left Lower Limb
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.
Fig. 3: Piecewise linear velocity parameterization model
Fig. 4: (a) Skeleton of lower limbs obtained from Vicon motion-capture data. (b) Simplified skeleton extracted from the Vicon motion-capture data.
Fig. 5: Snapshots of the suspended bipedal robot during experiments: (a) squatting; (b) walking.
Fig. 6: Comparison of reference and experimental joint angular trajectories for the suspended bipedal robot: (a) squatting; (b) walking.
Fig. 7: Experimental comparison of angular trajectories under the proposed method, MPC, and IPSO-PID for (a) squatting and (b) walking.
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
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 ( ∘ )
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)
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
This paper presents a multi-phase whole-body model predictive control (MPC) approach for bipedal walking, combining a detailed whole-body model in the near horizon with a simplified single-rigid-body model in the later prediction steps. This reduces computational complexity while retaining prediction capabilities. The resulting nonlinear optimal control problem is solved entirely within the general-purpose, off-the-shelf nonlinear MPC framework acados, using sequential quadratic programming (SQP). Given a contact schedule and a target walking speed, the controller optimizes joint torques without depending on preselected footstep locations. The controller is validated in MuJoCo simulation on the 18-DoF bipedal robot HyPer-2.
Franek Stark, Felix Wiebe, Shubham Vyas +2
Robotics Innovation Center at the German Research Center for Artificial Intelligence (DFKI), Bremen, Germany · University Bremen, Germany
In this letter, we present a hierarchical control framework that enables wheeled bipedal robots to perform planar object sliding tasks with their wheeled legs. The proposed approach formulates a nonlinear model predictive controller (NMPC) based on a reduced-order three rigid bodies (TRB) dynamical model that explicitly accounts for the hip roll degree of freedom and multiple wheel-environment contact modes, which is essential for lateral stepping and pedipulation tasks. Within this framework, the NMPC simultaneously regulates robot locomotion and interaction forces, allowing the robot to stably execute both rolling and object manipulation behaviors. A trajectory-optimization-based robot-object motion planner is developed to generate reference motions that incorporate stick-slip transitions in ground-object contact. Two representative pedipulation motions, namely scooting and lateral sliding, are validated through real-world hardware experiments, in which the robot successfully retrieves a 1 kg object from under a desk and slides a 4 kg object over a distance of 0.228 m via scooting.
Yue Qin, Yulun Zhuang, Zelin Shen +1
Department of Robotics, University of Michigan, Ann Arbor, MI 48109, USA
Bipedal loco-manipulation enables robots to interact with objects beyond the nominal workspace of their arms by coordinating locomotion and manipulation. Realizing this capability requires a low-level whole-body controller that translates task-level manipulation goals into coordinated arm and leg motions while maintaining balance. We present a unified whole-body controller trained with reinforcement learning that directly maps 6-DoF end-effector targets to coordinated actions for the bipedal base and robotic arm. Given only an end-effector target, the learned controller autonomously coordinates reaching, postural adaptation, and stepping without explicit base-velocity or footstep commands. A reward-gating strategy regulates the trade-offs among end-effector tracking, locomotion, and balance during training, while a temporal context estimator combines windowed Transformer encoding, recurrent GRU memory, and auxiliary dynamics prediction to extract dynamics-relevant information from observation history. Real-robot experiments demonstrate that the same controller supports reaching, postural adaptation, and stepping under commands from VR teleoperation, a learned diffusion policy, and scripted trajectories, providing a common end-effector interface for diverse manipulation tasks.