Stability in the sense of not falling is also difficult to encode in gait synthesis. Stable walking behavior is consequently generated by realizing the desired step size in the output structure and stabilizing the output via optimization-based controllers.
LIST OF TABLES
NOMENCLATURE
LIST OF ACRONYMS
BBF-QP Backstepping Barrier Function-Based Quadratic Programming TSC-QP Quadratic Programming Based on Task Space Control.
INTRODUCTION
Bipedal Robots
In other words, controlling the ankle angle to follow certain arbitrary trajectories can break the contact between the foot surface and the ground, making the robot no longer a fixed manipulator. That being said, the motion synthesis problem remains mathematically the same regardless of the type of robot operation.
State-of-the-art Approaches
The early development of HZD [134] focused on undersized planar walking, for which a monotonically increasing phase variable (e.g. angle of feet in stance) is defined as an internal clock. The desired trajectories of controlled states (outputs) are parameterized based on the phase variable.
Thesis Outline
Since the stabilization of the underactuation is implicitly checked after the planning (after the trajectory optimization), the underactuation is not directly processed in the form of active stabilization via re-planning. Instead of building on the HZD framework, we investigate directly on the underactivation, which will be explained in the final chapters of the approach.
PRELIMINARIES
Hybrid Dynamic Model of Walking
The evolution of the pre-collision state at the step level, i.e. the dynamics of the S2S robot, can be represented by. Feedback controllers can thus be synthesized based on S2S dynamics to stabilize SLIP operation.
Optimization-based Feedback Controller
In the last part of this thesis Eq. 2.12) will be directly referred to as the S2S dynamics of the biped walking robot. By examining the evolution of the top states, the S2S dynamics/return map of running can be easily obtained on the SLIP.
APPROACH ⊃ MODEL APPROXIMATION
- Hybrid Dynamics of Walking
- Step-to-step Dynamics
- Walking Synthesis
- S2S Dynamics based Stepping Stabilization
It creates continuous hybrid execution on the dynamic system and makes sure that the S2S dynamics of the robot exists. The S2S dynamics based step controller then stabilizes the walking of the robot close to the desired behavior.
APPROACH ⊃ ORBIT CHARACTERIZATION AND STABILIZATION
- Period-1 Orbit
- Period-2 Orbit
- Equivalent Characterization
- Orbit Stabilization
The initial states of the SSP of all periodic orbits for 𝑣𝑑 lie on the . Theorem 1.3 The orbital lines 𝑣 = ±𝜎2𝑝+𝑑2 characterize the P1 orbits when 𝑢L=𝑢R, yielding the final state of the SSP as.
APPLICATION ⊃ FIVE-LINKAGE WALKER AMBER
Robot Model
We assume that the impact is plastic: the mobile toe rests on the ground and the toe of the previous stance lifts off the ground. Both AMBER and five other bond bunnies [ 134 ] have been well studied in the HZD setting [ 50 , 134 ]. Here, we show the first application of the proposed H-LIP-based synthesis in Chapter 3 to generate walks on the five-link AMBER walker.
Walking Synthesis
Recall that walking in the HZD is described by periodic trajectories, where the activated trajectories are parameterized by the underactuated phase coordinate and then optimized via non-convex parameter optimization. Forward trajectory of the Swing Toe: The construction of the desired forward position of the Swing Toe is to realize the H-LIP based step controller, which calculates the desired step size based on the current horizontal COM status in the SSP. Output stabilization: Given the output construction, we apply the feedback controller in Chapter 2 to stabilize the outputs Y → 0.
Results and Conclusion
The horizontal dynamics of COM in the continuous phase is obviously not the same as H-LIP. Thus, the horizontal S2S dynamics of the walker is not the same as the S2S dynamics of the H-LIP. Since the stepping ensures that the state error is bounded in the perturbation invariant set, the gait of the walker can be realized to approximately follow the desired gait of the H-LIP.
APPLICATION ⊃ ACTUATED SLIP
- Model
- Indirect Approach via Trajectory Optimization
- Direct Approach via Backstepping Barrier Functions
- Conclusion
The complete expressions for the dynamics in the polar coordinates can be found in Appendix A. aSLIP is simulated from the initial state of the optimized stepping-in-place motion. In this chapter, we demonstrate the application of the H-LIP-based gait synthesis to the aSLIP model.
APPLICATION ⊃ 3D UNDERACTUATED BIPEDAL ROBOT CASSIE
- Robot Model
- Indirect Walking Synthesis
- Direct Walking Synthesis
- Conclusion
The desired orientations of the pelvis and swing foot are chosen to be constant. The speed of the robot is successfully controlled to the target speed by the H-LIP-based approach. The path of the leg swing is marked with a red (left leg) and blue (right leg) line.
APPLICATION ⊃ HARDWARE REALIZATION ON CASSIE
Hardware Implementation
So the current COM state of the robot[𝑝𝑡, 𝑣𝑡] in the SSP can be approximated by. Let ˜𝑣− be the approximate velocity of the COM of the robot at the pre-impact. Since the horizontal COM dynamics of the robot is close to the H-LIP dynamics, the velocity approximation works well (Fig.
Experiment Results
Variations in COM height and step duration change the S2S approximation of H-LIP (e.g., Eq. External thrusts directly perturb the S2S dynamics of the robot; ground variations vary. With thrust perturbations, the error state of 𝑦 can temporarily outside the invariant set 𝐸𝑦.
EXTENSION ⊃ GLOBAL POSITION CONTROL
S2S Dynamics with Global Position Control
Assuming that points are sampled on the time horizon[𝑡0, 𝑡0+𝑁 𝑇] to represent the trajectory of states, the cost function in tracking performance is. W are the step sizes available in each plane, which are linear functions of the states. 9.1 (b) shows an example of the optimized H-LIP walk with a torso in 3D for GPC.
Global Position Control on 3D-aSLIP
The optimized result on the H-LIP and the generated run on the aSLIP are shown in Fig. One thing to note is that the initial H-LIP state should be close to that of the horizontal state of the aSLIP. In the examples, we choose the H-LIP state to be initially identical to that of the aSLIP, indicating se0 = x˜ −x˜H-LIP = 0 ∈ 𝐸.
Global Position Control on Cassie
Tracking results: The proposed approach allows the robot to properly track all global trajectories in terms of position and velocity profiles (Fig. 9.4). Instead, as a proof-of-concept, we use the simulated walking speeds and rotational speeds of the robot as targets, which were tracked in [105] with a different controller on the robot. As the robot follows the commanded speeds, there is a slight deviation in position and direction as it executes the later segments of the shape.
EXTENSION ⊃ WALKING ON ROUGH TERRAIN
S2S Dynamics of H-LIP on Rough Terrain
In other words, the magnitude of the net acceleration is controllable by the leg forces. On rough terrain: The walking pattern of the H-LIP on general non-flat terrain is based on walking on slopes. Given a step sequence of H-LIP walking on rough terrain, walking is equivalent to walking on piecewise continuous slopes, as illustrated in Fig.
The uncertainty about the terrain height creates an uncertainty about the duration of the SSP: early strike produces a shorter duration and late strike produces a longer duration. We apply the H-LIP-based staircase by using the S2S dynamics of the H-LIP on rough terrain to approximate the S2S of the aSLIP model (see Fig. Again, we simply construct a smooth trajectory of the horizontal swing foot position to move from the previous step location to the desired location.
Results and Discussion
For even larger (unreal) noises, kinematic violation begins to occur, i.e., the foot hits the edges of the ground while walking. Similar to previous applications, the H-LIP step relies on the condition that the dynamics of aSLIP is close to that of H-LIP. The second is the time-based component of the vertical swing leg trajectory, which ensures that the SSP duration does not change if the ground noise𝛿 𝑧 is too large.
EXTENSION ⊃ TEMPLATE EMBEDDING ON HUMANOIDS
Dynamics Embedding with Contact Forces
The desired roll and pitch angles of the swivel foot are designed as zero points. Similar to the pelvic yaw angle, the desired yaw angle of the swivel base is designed to change when turning is required. The swivel foot position is built up smoothly from the start to the end position of the swivel foot in the SSP.
Motion Decoupling and Composition via ROMs
The optimization of the aSLIP model is also fast to solve due to its low dimensionality. In both cases, periodic walking is first composed of the periodic trajectories of aSLIP and H-LIP. The embedding of aSLIP that goes on the humanoid is the same as in the previous section (see Fig. 11.6).
Comparison
The MDC still relies on trajectory optimization on the ROMs to generate different periodic walking and transition walking motions. It is not highly efficient compared to the 3D-aSLIP embedding, but can generate walking with different step frequencies. This can be avoided by generating different periodic steps with different frequencies or changing the step frequency during step.
CONCLUSION
Contributions
Implications
In the presence of large tracking errors, the robot will not topple as long as the realized step sizes drive the horizontal COM state to the desired set.
Future Work
Instead, the kinematic feasibility is reflected on the choices of the desired gait of the H-LIP. In other words, the desired trajectories of the outputs are assumed to be traceable given the constraint of the motor design. Variations of the integration with step planning and continuous control via ankle actuation and momentum-based approaches can potentially open a large design space of the control synthesis for dynamic human walking in a challenging environment.
BIBLIOGRAPHY
Step-by-step dynamics approximation enables computationally efficient and fast control of legged robots. Dynamic walking over rough terrain by nonlinear predictive control of floating base inverted pendulum. Walking Control of Fully Actuated Robots Based on Biped Sliding Model.2012 IEEE International Conference on Robotics and Automation, pp.
EOM OF ASLIP
LEG SPRING ON CASSIE
Due to Cassie's specific leg design, the conformation appears mainly in the direction of the leg. Leg spring: Given the leg stiffness and the damping calculation, we have a non-linear spring in the leg. Since the internal dynamics of the leg are ignored, we see the activation of the leg length at the input of the leg.
UNCOMPRESSED OUTPUT ON CASSIE
BÉZIER POLYNOMIALS
