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.
Algorithm: Determination of the touchdown angle and the leg stiffness in solving problem Eq. (7)
Input:
The initial apex state S0
The target apex state Sd
The initial leg stiffness ks
Output:
The touchdown angle αTD
The leg stiffness kc and kd
1. Initialize the current leg stiffness
2. Compute the energy variation by using Eq. (9)
3. forαTD = π/4 to π/2 do
4. Compute the approximation of the leg length at BM
5.
6. Compute sub-maps , ,
7. Compute the A2RM
8. Compute the predicted apex state
9.
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
14. end algorithm
Tab.1
Fig.5
Fig.6
Fig.7
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
Parameter
Symbol
Value
Unit
Total mass
ms
19
kg
Leg length
r0
0.8
m
Leg stiffness
ks
3200
N/m
Tab.3
Fig.8
Fig.9
Fig.10
Fig.11
Fig.12
Fig.13
Fig.14
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
Fig.15
Fig.16
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