Model Predictive Control for

Legged Robots

Niraj Rathod

Computer Science and Systems Engineering, XXXIII Cycle

PhD defense

Prof. Alberto Bemporad

Advisors

IMT School for Advanced Studies Lucca, Italy

Dr. Michele Focchi

DLS Lab, Istituto Italiano di Tecnologia, Genova, and
University of Trento, Italy

Assoc. Prof. Mario Zanon

IMT School for Advanced Studies Lucca, Italy

 Why Legged Robots?

Agile

Strong

source:https://www.youtube.com/embed/tF4DML7FIWk?enablejsapi=1

source:https://youtu.be/pLsNs1ZS_TI

Cassie: https://news.engin.umich.edu/2017/09/latest-two-legged-walking-robot-arrives-at-michigan/

Hexaped: https://www.csiro.au/en/research/

HyQReal: https://dls.iit.it/web/dynamic-legged-systems/robots

sources:

Wheeled robot: https://www.leorover.tech/post/top-3-ways-to-set-a-robot-into-motion

Quadrotor: http://uosdesign.org/designshow2015/ni-myrio-autonomous-quadrotor/

sources:

Objective of this thesis

Robot

Task

Environment

Generate motion plan

Optimal control techniques

\mathbf{x}(t), \mathbf{u}(t)
\displaystyle{\min_{\mathbf{x}(t),\mathbf{u}(t)} \quad \int \ell\left(\mathbf{x}(t), \mathbf{u}(t)\right)}
\text{s.t.} \quad f\left(\mathbf{x}(t), \mathbf{u}(t)\right) = 0,
h\left(\mathbf{x}(t), \mathbf{u}(t)\right) \leq 0

Model predictive control with environment adaptation for legged locomotion, IEEE Access, vol. 9, pp. 145710-145727, 2021

Optimization-Based Reference Generator for Nonlinear Model Predictive Control of Legged Robots. Robotics 2023, 12(1), 6, 2023

?

Outline

Outline

Model Validation

Outline

Model Validation

Trajectory Optimization for Legged Locomotion

Outline

Model Validation

Trajectory Optimization for Legged Locomotion

Nonlinear Model Predictive Control

Outline

Model Validation

Nonlinear Model Predictive Control

Two-Stage Optimization

Trajectory Optimization for Legged Locomotion

Model validation

Model selection for optimal planning

High fidelity model

Rigid Body Dynamics

  • Accurate
  • Good for simulators and  offline Trajectory Optimization
\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{h}(\mathbf{q},\dot{\mathbf{q}})=\mathbf{S}^\top\bm{\tau}+\mathbf{J}_\mathrm{c}(\mathbf{q})^\top\mathbf{F}
\text{State}: \mathbf{q} \in \mathbb{SE}(3) \times \mathbb{R}^{n}
\text{Input}: \mathbf{u} = (\bm{\tau}, \mathbf{f}) \in  \mathbb{R}^{(n+3n_c)}
n \text{ is no. joints}
n_c\text{ is no. of end-effectors}
  • Might be unsuitable for real-time optimal planning

Model selection for optimal planning

High fidelity model

Rigid Body Dynamics

  • Accurate
  • Good for simulators and  offline Trajectory Optimization
\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{h}(\mathbf{q},\dot{\mathbf{q}})=\mathbf{S}^\top\bm{\tau}+\mathbf{J}_\mathrm{c}(\mathbf{q})^\top\mathbf{F}
\text{State}: \mathbf{q} \in \mathbb{SE}(3) \times \mathbb{R}^{n}
\text{Input}: \mathbf{u} = (\bm{\tau}, \mathbf{f}) \in  \mathbb{R}^{(n+3n_c)}
n \text{ is no. joints}
n_c\text{ is no. of end-effectors}
  • Might be unsuitable for real-time optimal planning

Approximate Model

Single Rigid Body Dynamics

m\dot{\mathbf{v}}_\mathrm{c}=m\mathbf{g}+\sum_{i=1}^{n_c}\mathbf{f}_i
\mathbf{I}_\mathrm{c}(\bm{\Phi})\,\dot{\bm{\omega}}+\bm{\omega} \times \mathbf{I}_\mathrm{c}(\bm{\Phi}) \bm{\omega} =\sum_{i=1}^{n_c} \mathbf{p}_{\mathrm{cf}_i} \times \mathbf{f}_i
\text{State}: \mathbf{x} = (\mathbf{p}_\mathrm{c}, \mathbf{v}_\mathrm{c}, \bm{\Psi}, \bm{\omega}) \in \mathbb{R}^{12}
\text{Input}: \mathbf{u} = \mathbf{f} \in  \mathbb{R}^{3n_c}
  • Includes linear and angular dynamics
  • Suitable for real-time optimal planning
  • Less accurate

Robot model

\begin{bmatrix} \dot{\mathbf{p}}_\mathrm{c}\\ \dot{\mathbf{v}}_\mathrm{c}\\ \dot{\bm{\Phi}}\\ \dot{\bm{\omega}} \end{bmatrix} = \begin{bmatrix} {\mathbf{v}}_\mathrm{c} \\ 1/m \sum_{i=1}^4 \mathbf{f}_i + \mathbf{g} \\ \mathbf{E}^{-1}(\bm{\Phi}) \bm{\omega} \\ -\mathbf{I}^{-1}_\mathrm{c}(\bm{\Phi})(\bm{\omega} \times \mathbf{I}_\mathrm{c}(\bm{\Phi}) \bm{\omega})+ \sum_{i=1}^4 \mathbf{I}^{-1}_\mathrm{c}(\bm{\Phi})(\mathbf{p}_{\mathrm{f}_i}-\mathbf{p}_c) \times \mathbf{f}_i \end{bmatrix}

Single Rigid Body Dynamics (SRBD)

m\dot{\mathbf{v}}_\mathrm{c}=m\mathbf{g}+\sum_{i=1}^{n_c}\mathbf{f}_i
\mathbf{I}_\mathrm{c}(\bm{\Phi})\,\dot{\bm{\omega}}+\bm{\omega} \times \mathbf{I}_\mathrm{c}(\bm{\Phi}) \bm{\omega} =\sum_{i=1}^{n_c} \mathbf{p}_{\mathrm{cf}_i} \times \mathbf{f}_i
\mathbf{x}_{k+1}=g_\mathrm{d}\left(\mathbf{x}_{k}, \mathbf{u}_{k}, \mathbf{p}_{k}\right)

Discrete-time state space model

\mathbf{x}_{k+1}=\mathbf{A}_k \mathbf{x}_k + \mathbf{B}_k \mathbf{u}_k +\mathbf{r}_k

Linear time-varying model

\dot{\mathbf{x}}(t)= g_\mathrm{c}(\mathbf{x}(t),\mathbf{u}(t),\mathbf{p}_{\mathrm{f}}(t))

Continuous-time state space model

\text{State}: \mathbf{x} = (\mathbf{p}_\mathrm{c}, \mathbf{v}_\mathrm{c}, \bm{\Psi}, \bm{\omega}) \in \mathbb{R}^{12}
\text{Input}: \mathbf{u} = \mathbf{f} \in  \mathbb{R}^{3n_c}
\mathbf{p}_\mathrm{c}: \text{CoM position},
\mathbf{E}(\bm{\Phi}): \text{Euler angle rates matrix}.
\bm{\Phi}: \text{Euler angle},
\bm{\omega}: \text{Angular velocity},
\mathbf{f}: \text{Ground reaction force},
\mathbf{p}_{\mathrm{f}}: \text{Foot positions},
\mathbf{I}_\mathrm{c}: \text{Inertia matrix},
\mathbf{v}_\mathrm{c}: \text{CoM velocity},
  • The SRBD model is nonlinear with the unstable dynamics
  • Therefore, it requires a stabilizing controller to simulate a nominal working condition

Closed-loop system validation

Why closed-loop system validation?

Closed-loop system validation

\dot{\mathbf{\tilde{\mathbf{x}}}}(t)=f(\mathbf{\tilde{x}}(t),\mathbf{\tilde{u}}(t))
\dot{\mathbf{x}}(t)=g(\mathbf{x}(t),\mathbf{u}(t))

Approximate model

For model validation:

\mathbf{x}(t) \simeq \mathbf{x}_\mathrm{sel}(t) \\ \mathbf{u}(t) \simeq \mathbf{u}_\mathrm{sel}(t)
\dot{\mathbf{\hat{\mathbf{x}}}}(t)=f(\mathbf{\hat{x}}(t),\mathbf{\hat{u}}(t))
\mathbf{\hat{u}}(t) = \hat{\kappa}(\mathbf{\hat{x}}(t), \mathbf{\hat{x}}^\mathrm{ref}(t))
\mathbf{u}(t) = \kappa(\mathbf{x}(t), \mathbf{x}^\mathrm{ref}(t))
\dot{\mathbf{x}}_\mathrm{sel}(t)=f_\mathrm{sel}(\mathbf{x}_\mathrm{sel}(t),\mathbf{u}_\mathrm{sel}(t), \\ \qquad \qquad\mathbf{\hat{x}}(t),\mathbf{\hat{u}}(t))
\mathbf{u}_\mathrm{sel}(t) = \kappa(\mathbf{x}_\mathrm{sel}(t), \mathbf{x}^\mathrm{ref}_\mathrm{sel}(t))
\mathbf{x}_\mathrm{ref} \rightarrowtail \mathbf{x}_\mathrm{sel}

validated by

g

Validation results

Validation data: HyQ crawling with the body pitch and yaw

Pitch

yaw

Ground reaction forces for pitch validation

Trajectory Optimization for Legged Locomotion

Trajectory Optimization: LTVOpt

\displaystyle{\min_{\mathbf{x}^\mathrm{p},\mathbf{u}^\mathrm{p}}}\quad \sum_{k=0}^{N-1} \parallel \mathbf{x}_k -\mathbf{x}_k^\mathrm{ref}\parallel_{\mathbf{W}_\mathrm{x}}^2 + \parallel \mathbf{u}_k -\mathbf{u}_k^\mathrm{ref}\parallel_{\mathbf{W}_\mathrm{u}}^2 \\ + \parallel \mathbf{x}_N -\mathbf{x}_N^\mathrm{ref}\parallel_{\mathbf{W}_\mathrm{x_N}}^2 \\
\text{s.t.} \quad \mathbf{x}_{0}=\mathbf{\hat{x}}_0, \\ %&\mathbf{x}_{k+1}=\mathbf{A}_k\mathbf{x}_{k} + \mathbf{B}_k \mathbf{u}_{k}+ \mathbf{r}_{k},&& k\in\mathbb{I}_0^{N-1},\\ % &h\left(\mathbf{x}_{k}, \mathbf{u}_{k}, \mathbf{a}_{k}\right) \leq 0,&& k\in\mathbb{I}_0^{N-1}, \label{eq:ltv_inequality} %&\mathbf{C}_k \mathbf{x}_k+\mathbf{D}_k \mathbf{u}_k+\mathbf{h}_k \geq 0, && k\in\mathbb{I}_0^{N-1}
\mathbf{x}_{k+1}=\mathbf{A}_k\mathbf{x}_{k} + \mathbf{B}_k \mathbf{u}_{k}+ \mathbf{r}_{k},\qquad k\in\mathbb{I}_0^{N-1} %&\mathbf{C}_k \mathbf{x}_k+\mathbf{D}_k \mathbf{u}_k+\mathbf{h}_k \geq 0, && k\in\mathbb{I}_0^{N-1}
\mathbf{C}_k \mathbf{x}_k+\mathbf{D}_k \mathbf{u}_k+\mathbf{h}_k \geq 0, \quad \, \qquad k\in\mathbb{I}_0^{N-1} %&\mathbf{C}_k \mathbf{x}_k+\mathbf{D}_k \mathbf{u}_k+\mathbf{h}_k \geq 0, && k\in\mathbb{I}_0^{N-1}

LTV form of the SRBD model

Friction cone and unilateral constraints

Reference \(\mathbf{x}^\mathrm{ref}\) and \(\mathbf{u}^\mathrm{ref}\) generation from heuristic

State \(\mathbf{x}\) and Input \(\mathbf{u}\) tracking term

Uses references (often dynamically infeasible) as linearization trajectory to obtain LTV model

The solution of the LTVOpt may be dynamically infeasible

Initial condition

So the proposed solution is ...

Trajectory Optimization: SQPOpt

\displaystyle{\min_{\Delta \mathbf{x}, \Delta \mathbf{u}}}\quad \sum_{k=0}^{N-1} \frac{1}{2} \left [\begin{array}{c}\Delta \mathbf{x}_k \\ \Delta \mathbf{u}_k\end{array}\right]^\top \mathbf{H}_k \left[ \begin{array}{c}\Delta \mathbf{x}_k \\ \Delta \mathbf{u}_k\end{array} \right]+ \mathbf{J}_k^\top \left[ \begin{array}{c}\Delta \mathbf{x}_k \\ \Delta \mathbf{u}_k\end{array} \right]
\mathrm{s.t.} \quad \Delta \mathbf{x}_0=\hat{\mathbf{x}}_0- \mathbf{x}_0^{\mathrm{L}}, %&\Delta \mathbf{x}_{k+1}=\mathbf{A}_k \Delta \mathbf{x}_k+\mathbf{B}_k \Delta \mathbf{u}_k+\mathbf{r}_k, \label{eq:qp_equality}\\ %&\mathbf{C}_k \Delta \mathbf{x}_k+\mathbf{D}_k \Delta \mathbf{u}_k+\mathbf{h}_k \geq 0, \label{eq:qp_inequality}
  • Gives dynamically feasible solution if converged
  • LTV form of the SRBD model
  • Quadratic cost:  State \(\mathbf{x}\) and Input \(\mathbf{u}\) tracking term
\Delta \mathbf{x}_{k+1}=\mathbf{A}_k \Delta \mathbf{x}_k+\mathbf{B}_k \Delta \mathbf{u}_k+\mathbf{r}_k, \qquad k\in\mathbb{I}_0^{N-1}\\ %&\mathbf{C}_k \Delta \mathbf{x}_k+\mathbf{D}_k \Delta \mathbf{u}_k+\mathbf{h}_k \geq 0, \label{eq:qp_inequality}
\mathbf{C}_k \Delta \mathbf{x}_k+\mathbf{D}_k \Delta \mathbf{u}_k+\mathbf{h}_k \geq 0 \,\,\, \qquad \qquad k\in\mathbb{I}_0^{N-1}
  • Usually solves more than one QP in a single run therefore can be computationally demanding
  • Friction cone and unilateral constraints

Back Tracking Algorithm

Results: LTVOpt and SQPOpt

CoM position

CoM velocity

  • Prediction horizon: 4 s (100 nodes)
  • Sampling time 40 ms
  • Solver: Gurobi
  • Solution time:
    • LTVOpt : 200 ms 
    • SQPOpt : 1-1.2 s 

Implementation details:

Results: LTVOpt and SQPOpt

Ground reaction forces: normal components

Orientation

Angular velocity

Nonlinear Model Predictive Control

Why NMPC?

  • Many successful MPC implementation [1], [2], [3] examples in legged locomotion
  • Some MPC implementation either neglect angular dynamics [4] or linearize it [5] that is not good for large variation from the horizontal orientation
  • We would like to facilitate locomotion that involves considerable base orientation
  • NMPC unlike linear MPC allows one to include nonlinear dynamics, non-quadratic cost and nonlinear inequalities

[1] J. Koenemann, et al. Whole-body model-predictive control applied to the HRP-2 humanoid. In IEEE/RSJ Int. Conf. Intell. Robot. Syst., pages 3346–3351, 2015.

[2 ] M. Neunert, et al. Whole-Body Nonlinear Model Predictive Control Through Contacts for Quadrupeds. IEEE Robot. Autom. Lett., 3(3):1458–1465,2018.

[3] R. Grandia, et al. Feedback mpc for torque controlled legged robots. In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 4730–4737, 2019.

[4] A. Herdt, et al. Online walking motion generation with automatic footstep placement. Advanced Robotics, 24(5-6):719–737, 2010.

[5] J. Di Carlo, et al. Dynamic locomotion in the MIT cheetah 3 through convex model-predictive control. In 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 1–9,2018. 

Towards Real-Time Iteration scheme

  • Since LTVOpt solves one QP at each call it is faster than SQPOpt
  • LTVOpt uses reference as linearization trajectory whereas SQPOpt warm starts after the first iteration
  • SQPOpt solves until convergence thus can provide feasible solution
  • To solve only one QP online
  • To warm start from the previous solution to evaluate the sensitivities in the next call of the algorithm

We use Real-Time Iteration (RTI) scheme for our NMPC.

For online implementation it is desirable:

Locomotion framework

NMPC runs at \( 25\, \mathrm{Hz} \)

WBC runs at \( 250\, \mathrm{Hz} \)

State estimator runs at \( 500\, \mathrm{Hz} \)

Joint Space PD runs at \( 1\, \mathrm{kHz} \)

Grid map runs at \( 30\, \mathrm{Hz} \)

NMPC formulation

State \(\mathbf{x}_k \in \mathbb{R}^{12} \) 

vector of the robot's CoM position, CoM velocity, base orientation, and angular velocity

Control input \(\mathbf{u}_k \in \mathbb{R}^{12} \)

vector of ground reaction forces

 vector of foot locations and contact status

Model parameters \(\mathbf{a}_k \in \mathbb{R}^{16} \)

Decision variables:

Predicted state \(\mathbf{x}^\mathrm{p}=\{\mathbf{x}_{0}, \ldots, \mathbf{x}_N\}\)

Control inputs \(\mathbf{u}^\mathrm{p}=\{\mathbf{u}_{0}, \ldots, \mathbf{u}_{N-1}\}\)

 Nonlinear Programming problem

\displaystyle{\min_{\mathbf{x}^\mathrm{p},\mathbf{u}^\mathrm{p}}} \quad \sum_{k=0}^{N-1} \ell\left(\mathbf{x}_{k}, \mathbf{u}_{k}, \mathbf{a}_{k}\right)+\ell_\mathrm{T}\left(\mathbf{x}_{N}\right)
\text{s.t.} \quad \mathbf{x}_{0}=\boldsymbol{\hat{\mathbf{x}}_{0}},
\mathbf{x}_{k+1}=f\left(\mathbf{x}_{k}, \mathbf{u}_{k}, \mathbf{a}_{k}\right), \quad k\in\mathbb{I}_0^{N-1},
h\left(\mathbf{x}_{k}, \mathbf{u}_{k}, \mathbf{a}_{k}\right) \leq 0, \qquad k\in\mathbb{I}_0^{N-1}

Cost function

  • \(\ell_\mathrm{t}\)  is tracking cost on \(\mathbf{x}_k \in \mathbb{R}^{12} \) and control inputs \(\mathbf{u}_k \in \mathbb{R}^{12} \)
\ell\left(\mathbf{x}_{k}, \mathbf{u}_{k},\mathbf{a}_k\right)= \,\ell_\mathrm{t} + \ell_\mathrm{m} + \ell_\mathrm{r},
\ell_\mathrm{t} =\parallel \mathbf{x}_k-\mathbf{x}^{\mathrm{ref}}_k\parallel_{\mathbf{Q}}^2+ \parallel \mathbf{u}_k-\mathbf{u}^{\mathrm{ref}}_k\parallel_{\mathbf{R}}^2,
\ell_\mathrm{m} = \parallel {}_\mathcal{C}\mathbf{p}_{\mathrm{hf}_{k}}-{}_\mathcal{C}\mathbf{p}_{\mathrm{hf}_{k}}^{\mathrm{ref}}\parallel_{\mathbf{M}}^2,
\ell_\mathrm{r} = \,\rho\parallel {{}_\mathcal{K}\mathbf{u}_k}\parallel_{\mathbf{P}}^2
  • \(\ell_\mathrm{r}\) is the regularization term for the control inputs to keep contact normal as close to the center of the friction cone
  • \(\ell_\mathrm{m}\) is the mobility cost that penalizes hip-to-foot distance from the reference

SRBD Model for NMPC

\begin{bmatrix} \dot{\mathbf{p}}_\mathrm{c}\\ \dot{\mathbf{v}}_\mathrm{c}\\ \dot{\bm{\Phi}}\\ {}_\mathcal{C}\dot{\bm{\omega}} \end{bmatrix} = \begin{bmatrix} {\mathbf{v}}_\mathrm{c} \\ 1/m \sum_{i=1}^4 {\color{red}{\delta_i}} \, {}_\mathcal{C}\mathbf{f}_i + \mathbf{g} \\ \mathbf{E}^{-1}(\bm{\Phi}) {}_\mathcal{C}\bm{\omega} \\ -{\color{blue} {}_\mathcal{C}\mathbf{I}_\mathrm{c}}^{-1}({}_\mathcal{C}\bm{\omega} \times {\color{blue} {}_\mathcal{C}\mathbf{I}_\mathrm{c}}\, {}_\mathcal{C}\bm{\omega})+ \sum_{i=1}^4 {\color{red}{\delta_i}} \, {\color{blue} {}_\mathcal{C}\mathbf{I}_\mathrm{c}}^{-1} {}_\mathcal{C}\mathbf{p}_{\mathrm{cf},i} \times {}_\mathcal{C} \mathbf{f}_i \end{bmatrix}
  • Defined in the CoM frame, so constant inertia
  • Binary parameters \(\delta_i = \{0, 1\}\) to define whether foot \(i\) is in contact with the ground
m\dot{\mathbf{v}}_\mathrm{c}=m\mathbf{g}+\sum_{i=1}^{n_c}\mathbf{f}_i
\mathbf{I}_\mathrm{c}(\bm{\Phi})\,\dot{\bm{\omega}}+\bm{\omega} \times \mathbf{I}_\mathrm{c}(\bm{\Phi}) \bm{\omega} =\sum_{i=1}^{n_c} \mathbf{p}_{\mathrm{cf}_i} \times \mathbf{f}_i
m\dot{\mathbf{v}}_\mathrm{c}=m\mathbf{g}+\sum_{i=1}^{4}{\color{red}{\delta_i}}\mathbf{f}_i
{\color{blue} {}_\mathcal{C}\mathbf{I}_\mathrm{c}}\,{}_\mathcal{C}\dot{\bm{\omega}}+{}_\mathcal{C}\bm{\omega} \times {\color{blue} {}_\mathcal{C}\mathbf{I}_\mathrm{c}}\, {}_\mathcal{C}\bm{\omega} =\sum_{i=1}^{4} {\color{red}{\delta_i}} \, {}_\mathcal{C}\mathbf{p}_{\mathrm{cf},i} \times {}_\mathcal{C}\mathbf{f}_i

Defined in the CoM frame

Included  \(\delta_i\)

Mobility

l_\mathrm{m} = \beta \frac{V}{\bar{V}} - \gamma \frac{E}{\bar{E}}
  • Parameters \(\beta\) and \(\gamma\) are introduced to find the best trade-off between the volume and eccentricity
  • Mobility factor \(l_\mathrm{m}\) can be evaluated inside the workspace of each leg
  • This analysis outputs the hip-to-foot distance belonging to the highest mobility that is used as a reference for the mobility cost inside the NMPC

Simulation: NMPC with mobility

Simulation: NMPC with pallet crossing

Simulation: NMPC with force robustness

Experiment: NMPC omni-directional walk

Experiments: NMPC with traversing pallets

Static pallet

Repositioned pallet

Experiments: Trotting in a straight line

Two-Stage Optimization

Optimization-based reference generator

  • Provides physically informed reference trajectory to the NMPC
  • Computes the foothold heuristically coherent with the CoM position 

Why is it needed?

  • Takes care of the underactuation of statically unstable gaits like trot
  • Allows to cope with the external disturbances

Optimization-based reference generator

\mathbf{x}^\mathrm{ref}
\mathbf{u}^\mathrm{ref}

NMPC

\mathbf{x}^\mathrm{g}_\mathrm{c}, \mathbf{w}^\mathrm{g} = \argmin \text{LIPopt}(\mathbf{x}_\mathrm{c,0}^\mathrm{act}, \mathbf{p}_\mathrm{f}, \bm{\delta})

LIP model optimization

\mathbf{x}^\mathrm{g}_\mathrm{c} = (\mathbf{p}_\mathrm{c,x}, \mathbf{p}_\mathrm{c,y}, \mathbf{v}_\mathrm{c,x}, \mathbf{v}_\mathrm{c,y}) \text{ is CoM X-Y pos. \& vel.}
\mathbf{w}^\mathrm{g} = (\mathbf{w}_\mathrm{c,x}, \mathbf{w}_\mathrm{c,y}) \text{ is ZMP X-Y position}
\mathbf{u}^\mathrm{qp} = \argmin \text{QPMap}(\mathbf{x}^\mathrm{g}_\mathrm{c}, \mathbf{w}^\mathrm{g})

QP mapping

\mathbf{u}^\mathrm{qp} \text{ is Ground Reaction Forces}
\mathbf{x}^\mathrm{g}_\mathrm{c}, \mathbf{w}^\mathrm{g}

+

Remaining state from heuristics

Simulation: pace gait

Experiment: trotting in a straight line

Experiment: recovery from external pushes

Summary and future directions

Summary

  • Model validation of the SRBD model
  • Introduced two Trajectory optimization schemes
  • Nonlinear Model Predictive Control with leg mobility
  • Two-Stage Optimization: NMPC with optimal reference generator
  1. LTVOpt
  2. SQPOpt

Future directions

  • Auto-tuning algorithm to tune the NMPC weights
  • Extend our NMPC formulation to optimize the step timing and foot locations
  • NMPC with nonuniform grid discretization method
  • A comparative study of linear MPC, LTV-MPC, and NMPC for legged locomotion

Publications

N. Rathod, A. Bratta, M. Focchi, M. Zanon, O. Villarreal, C. Semini, A. Bemporad, “Model Predictive Control with Environment Adaptation for Legged Locomotion”, IEEE Access, vol. 9, pp. 145710-145727, 2021.

A. Bratta, M. Focchi, N. Rathod, and C. Semini, “Optimization-Based Reference Generator for Nonlinear Model Predictive Control of Legged Robots”, Robotics 2023, 12(1), 6, 2023.

Journal papers

A. Bratta, N. Rathod, M. Zanon, O. Villarreal, A. Bemporad, C. Semini, M. Focchi, “Towards a Nonlinear Model Predictive Control for Quadrupedal Locomotion on Rough Terrain”, Italian Institute of Robotics and Intelligent Machines (I-RIM) Conference, 3rd edition, Rome, Italy, 2021.

Conference proceedings and workshops

N. Rathod, M. Focchi, M. Zanon, A. Bemporad, “Optimal control based replanning for Quadruped robots”, Numerical Optimization for Online Multi Contact Planning and Control, Robotics: Science and Systems, Freiburg, Germany, 2019.

N. Rathod, A. Bratta, M. Focchi, M. Zanon, O. Villarreal, C. Semini, A. Bemporad, “Real-time MPC with Mobility enhanced Feature for Legged Locomotion”, Towards Real-World Deployment of Legged Robots, ICRA, Xi’an China, 2021.

[2021 IEEE Access Best Video Award (Part 2)]

[Finalist for the Best Student Paper Award]

Collaborators

Claudio Semini

Head of the Dynamic Legged Systems (DLS) research line
Senior Researcher Tenured - Principal Investigator

Istituto Italiano di Tecnologia, Genova

Angelo Bratta

Postdoctoral Researcher at Dynamic Legged Systems lab, Istituto Italiano di Tecnologia, Genova

Octavio Villarreal

Senior Robotics Research Controls Engineer Dyson

City of Bristol, England, United Kingdom

Thanks

MPC for Legged Locomotion

By nirajrathod806

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MPC for Legged Locomotion

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