Fundamentos para el diseño de un PSA

1.4.1 Main Parts of the Dissertation

In this dissertation, to present how we achieved those research objectives, we split those studies into three main parts: i) Quadratic program-based walking controller de- sign, ii) Trajectory optimization, and iii) Capture point-based method for human motion analysis. The scopes and the capabilities of the algorithms developed in those parts are summarized in Table 1.1.

Table 1.1: The summary of the scopes and capabilities of the algorithms developed in different parts in this dissertation.

Parts QP-based Trajectory Capture point

controller optimization -based analysis

Model RBD + LIPM RBD LIPM

Torque saturation _{3 3 7}

Balance criteria 3 7 3

Feedback control 3 7 7

Constrained optimal control _{3 3 7}

Foot-rolling motion 7 3 7

Step length estimation 7 7 3

Human motion analysis _{7 7 3}

In the following subsections, we will introduce each of the three main parts in the dis- sertation, including the corresponding chapters, their relations to each research objective and our contributions.

1.4.2 Quadratic Program-based Walking Controller Design

The main purpose of this part is to achieve the research objective R1, which is de- scribed in Chapter 2. There are several potential issues about the model mismatch when applying MPC and the low-level control in sequence. First, for MPC with the LIPM (COM planning), although the dynamic balance (i.e. ZMP constraints) over the horizon can be imposed into the constrained MPC, the planned COM is based on simplified model there- fore may not reflect the full dynamics of the bipedal robot. Second, for the nonlinear low-level control, although one can adopt a constrained optimal control to track the de- sired trajectory and impose the ZMP for the current time step, there is no guarantee that the state won’t enter the region where the ZMP constraint in the next time step will be vi- olated. To address those issues, by leveraging the fact that both the constrained MPC and constrained nonlinear control can be expressed as quadratic programs (QPs), we propose a QP-based controller design to combine both QPs into a single framework, with a synthesis equality constraint to equal the COM accelerations derived from the LIPM and RBD. In this way, the unified QP will simultaneously generate the COM motion (which satisfies the ZMP constraint over the horizon, and is with the feedback from the nonlinear RBD) and the control input (which can track alone the generated COM under torque saturation, dynamic balance, and Lyapunov stability constraints for the current time step).

1.4.3 Trajectory Optimization

For achieving research objectives (R2 – R5), Chapters 3 to 5 are the studies to explore different trajectory optimization algorithms with direct collocation framework.

optimization under terrain uncertainties, which is described in Chapter 3. In this work, by utilizing the structure of direct collocation method, the last few collocation points are used to sample the walking trajectory under terrain uncertainties – in this case, instead of sampling the walking trajectory with different step height similar to the works in [32, 33] (which complicate the trajectory optimization problem), we sample the walking trajectory with different step-time and design a robust cost function to improve the gait robustness without complicating the collocation framework.

To generate energetically efficient gait with multiple domains towards human-like mo- tion, both Trajectory optimization through contact [30] and Hybrid Zero Dynamics (HZD) gait optimization [24] are the main methods we use to develop our works further. In Chap- ter 4, we modify the optimization through contact to generate human-like level walking for objectives R3 – R4. With more accurate transcription (using Hermite-Simpson method), we compared the generated level walking with different contact constraints, and we also compared the optimization results to the human data. In Chapter 5, to reduce the sensi- tivity of the optimization to the randomized initial guess, the HZD gait optimization is implemented, which covers the objectives R3 – R5. With the modified contact constraints, the optimization can be generally used on different terrains include flat ground, different slopes and stairs. To analyze the sensitivity of the HZD gait optimization to the initial guess, the optimization performance with the randomized initial guesses under different terrain profiles are also evaluated and compared. The details of the trajectory optimization algorithms developed in this dissertation are summarized in Table 1.2.

1.4.4 Capture Point-Based Method for Human Motion Analysis

The works of this part is described in Chapter 6. The CP-based step estimation for step- recovery (objective R6) was studied by comparing the estimated step location to the human experimental results from the literature [4, 6] and the estimation from the optimization

Table 1.2: The capabilities of the algorithms of trajectory optimization developed in this dissertation. All the methods we developed use the Hermite-Simpson Method (HSM) as the transcription method.

Topics Robust trajectory Trajectory optimization HZD gait optimization through contact optimization

Model RBD RBD RBD Transcription method HSM HSM HSM Robustness under _{3 7 7} terrain uncertainties Multiple (contact) _{7 3 3} domains Contact sequence 7 3 7 generation

Sensitivity to initial guess medium high low

Level walking _{7 3 3}

Slope walking _{3 7 3}

Stair walking 7 7 3

using the simulation on a simplified model with MPC [5]. For the case of objectives R7 and R8, we compared the CP-based step estimation to the simulation data of robot walkers and the experimental data of human subjects. The results indicate that capture point can provide good estimations for human walking and walking with mild-slip (which is defined as the peak heel velocity (PHV) is < 1.44m/s [34]).