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Frontiers of Mechanical Engineering

Front. Mech. Eng.    2020, Vol. 15 Issue (2) : 193-208
Sagittal SLIP-anchored task space control for a monopode robot traversing irregular terrain
Haitao YU(), Haibo GAO, Liang DING, Zongquan DENG
State Key Laboratory of Robotics and Systems, Harbin Institute of Technology, Harbin 150001, China
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As a well-explored template that captures the essential dynamical behaviors of legged locomotion on sagittal plane, the spring-loaded inverted pendulum (SLIP) model has been extensively employed in both biomechanical study and robotics research. Aiming at fully leveraging the merits of the SLIP model to generate the adaptive trajectories of the center of mass (CoM) with maneuverability, this study presents a novel two-layered sagittal SLIP-anchored (SSA) task space control for a monopode robot to deal with terrain irregularity. This work begins with an analytical investigation of sagittal SLIP dynamics by deriving an approximate solution with satisfactory apex prediction accuracy, and a two-layered SSA task space controller is subsequently developed for the monopode robot. The higher layer employs an analytical approximate representation of the sagittal SLIP model to form a deadbeat controller, which generates an adaptive reference trajectory for the CoM. The lower layer enforces the monopode robot to reproduce a generated CoM movement by using a task space controller to transfer the reference CoM commands into joint torques of the multi-degree of freedom monopode robot. Consequently, an adaptive hopping behavior is exhibited by the robot when traversing irregular terrain. Simulation results have demonstrated the effectiveness of the proposed method.

Keywords legged robots      spring-loaded inverted pendulum      task space control      apex return map      deadbeat control      irregular terrain negotiation     
Corresponding Author(s): Haitao YU   
Just Accepted Date: 11 February 2020   Online First Date: 09 March 2020    Issue Date: 25 May 2020
 Cite this article:   
Haitao YU,Haibo GAO,Liang DING, et al. Sagittal SLIP-anchored task space control for a monopode robot traversing irregular terrain[J]. Front. Mech. Eng., 2020, 15(2): 193-208.
Fig.1  Schematic of the sagittal SLIP-anchored task space control architecture. SLIP: Spring-loaded inverted pendulum.
Fig.2  Illustration of the sagittal SLIP model. (a) Coordinates and variable definitions; (b) entire gait cycle, including flight and stance sub-phase.
Fig.3  Composition of ARM, including the three sub-maps of Pfd, Pst, and Pfu.
Fig.4  Schematic of the variable stiffness policy with piecewise constant profile for the compression and decompression phases. (a) Division of the stance phase, with the instant stiffness variation at the bottom defined as the maximum leg compression; (b) variable stiffness spring with piecewise-constant leg stiffness.
Algorithm: Determination of the touchdown angle and the leg stiffness in solving problem Eq. (7)
The initial apex state S0
The target apex state Sd
The initial leg stiffness ks
The touchdown angle αTD
The leg stiffness kc and kd
1. Initialize the current leg stiffness kcks
2. Compute the energy variation ΔEby using Eq. (9)
3. for αTD = π/4 to π/2 do
4. Compute the approximation of the leg length at BM r˜BM
5. kdkc +2ΔE/ (r r˜BM)2
6. Compute sub-maps Pfd, P˜st, Pfu
7. Compute the A2RM P ˜ P fdP˜stPfu
8. Compute the predicted apex state S˜ n+1 P˜(S0, (αTD ,kc, kd) T)
9. αTD=arg min Sd P˜(S0 ,( αTD,k c,kd)T)
10. end for
11. redo Steps 3 and 4
12. return αTD, kc, and kd
13 Update the leg stiffness for the coming compression sub-phase with kskd
14. end algorithm
Fig.5  Hopping with constant absolute altitude when traversing irregular terrains in the SLIP model. Colored curves represent CoM trajectories of different terrain irregularities. y0 and yd represents the initial and target hopping height, respectively.
Fig.6  Rigid model of the monopode robot. (a) Leg configuration and relevant parameters; (b) entire hopping gait cycle with sub-phase division. AoT αTD and virtual equivalent leg req are the corresponding parameters used in the SLIP model.
Fig.7  FSM for monopode robot hopping.
Parameter Symbol Value Unit
Upper body mass mb 12 kg
Shank mass m1 3.5 kg
Thigh mass m2 3.5 kg
Shank inertia J1 0.08 kg·m2
Thigh inertia J2 0.08 kg·m2
Shank length l1 0.5 m
Thigh length l2 0.5 m
Shank CoM length lC1 0.25 m
Thigh CoM length lC2 0.25 m
Tab.2  Model parameters of the monopode robot in the simulation
Parameter Symbol Value Unit
Total mass ms 19 kg
Leg length r0 0.8 m
Leg stiffness ks 3200 N/m
Tab.3  Model parameters of the sagittal SLIP model in the simulation
Fig.8  Snapshots of the CoM trajectory of the monopode robot to represent periodic hopping on a flat surface.
Fig.9  Simulation results of selected variables for the monopode robot on constant apex height tracking. (a) Evolution of joint angles q1 and q2; (b) apex height ya and AoT αTD from stride to stride.
Fig.10  Snapshots of the CoM trajectory of the monopode robot executing target apex tracking on a flat surface.
Fig.11  Simulation results of selected variables of the monopode robot on variable apex state tracking. (a) Evolution of joint angles q1 and q2; (b) apex height and velocity from stride to stride.
Fig.12  Snapshots of the CoM trajectory of the monopode robot traversing an irregular terrain.
Fig.13  Simulated horizontal velocity and vertical height of CoM of the monopode robot.
Fig.14  Phase portrait of joint angles q1 and q2 of the robot traversing an irregular terrain. Arrows represent the evolution of the selected variables over time.
Comparison items Traditional SLIP controller [10] Proposed deadbeat controller
System representation Nonlinear differential equations Analytical approximations
Control input AoT AoT and leg stiffness
Control policy Fixed AoT Adjustable AoT and leg stiffness
Steering duration Only flight phase Both flight and stance phase
System energy Conservation Adding/removing energy
Steerable apex state Height or velocity height and velocity (independent)
Period of apex steering Asymptotically Only within a one-gait cycle
Terrain adaptability Flat ground Flat and uneven ground
Practical implementation Fourth-order Runge–Kutta solver Direct coding
Tab.4  General feature comparison between the traditional SLIP controller and the proposed controller
Fig.15  Comparative simulation results of the monopode robot equipped with the traditional and proposed controllers. Shaded areas represent convergence duration for each case.
Fig.16  ARM of the SLIP model equipped with traditional fixed AoT policy. Shaded areas represent the unfeasible region in which stumbling occurs with insufficient initial releasing height ya(i)<r0sinαTD. Stable and unstable fixed points are colored green and red, respectively. The partial view shows the convergence process after initial release at AoT αTD = 64°.
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