Simultaneous Contact, Gait and Motion Planning for Robust Multi-Legged Locomotion via Mixed-Integer Convex Optimization

Bernardo Aceituno,

Carlos Mastalli,

Hongkai Dai,

Michele Focchi,

Andreea Radulescu,

Darwin G. Caldwell,

Jose Cappelletto,

Juan C. Grieco,

G. Fernandez-Lopez,

Claudio Semini

Centroidal Dynamic Model

  • Non-coplanar stability through friction cone constraints
  • The centroidal dynamics describes the system's evolution
  • Integer variables model the contacts and gait transitions
  • Approximate the robot's joint torque limits

We represent the time evolution of the system using a centroidal dynamics model [1]

\begin{bmatrix} m \ddot{\mathbf{r}} \\ \dot{\mathbf{k}} \end{bmatrix} = \begin{bmatrix} m \mathbf{g} + \sum_{l = 1}^{n_l} \mathbf{\lambda}_{l} \\ \sum_{l = 1}^{n_l} (\mathbf{p}_{l} - \mathbf{r}) \times \mathbf{\lambda}_{l} \end{bmatrix}
[mr¨k˙]=[mg+∑l=1nlλl∑l=1nl(pl−r)×λl]\begin{bmatrix} m \ddot{\mathbf{r}} \\ \dot{\mathbf{k}} \end{bmatrix} = \begin{bmatrix} m \mathbf{g} + \sum_{l = 1}^{n_l} \mathbf{\lambda}_{l} \\ \sum_{l = 1}^{n_l} (\mathbf{p}_{l} - \mathbf{r}) \times \mathbf{\lambda}_{l} \end{bmatrix}

where

  • CoM position
  • Angular Momentum
  • Contact force of end-effector
  • Position of end-effector

Centroidal Dynamic Model

\mathbf{r}
r\mathbf{r}
\mathbf{k}
k\mathbf{k}
\mathbf{\lambda}_l
λl\mathbf{\lambda}_l
\mathbf{p}_l
pl\mathbf{p}_l

[1] Orin et al., "Centroidal dynamics of a humanoid robot", 2013

We represent the time evolution of the system using a centroidal dynamics model

\begin{bmatrix} m \ddot{\mathbf{r}} \\ \dot{\mathbf{k}} \end{bmatrix} = \begin{bmatrix} m \mathbf{g} + \sum_{l = 1}^{n_l} \mathbf{\lambda}_{l} \\ \sum_{l = 1}^{n_l} (\mathbf{p}_{l} - \mathbf{r}) \times \mathbf{\lambda}_{l} \end{bmatrix}
[mr¨k˙]=[mg+∑l=1nlλl∑l=1nl(pl−r)×λl]\begin{bmatrix} m \ddot{\mathbf{r}} \\ \dot{\mathbf{k}} \end{bmatrix} = \begin{bmatrix} m \mathbf{g} + \sum_{l = 1}^{n_l} \mathbf{\lambda}_{l} \\ \sum_{l = 1}^{n_l} (\mathbf{p}_{l} - \mathbf{r}) \times \mathbf{\lambda}_{l} \end{bmatrix}

The non-linear term                        can be described as bilinear function [2], and it can be decomposed as:

Centroidal Dynamic Model

(\mathbf{p}_l - \mathbf{r}) \times \mathbf{\lambda}_l
(pl−r)×λl(\mathbf{p}_l - \mathbf{r}) \times \mathbf{\lambda}_l
\mathbf{ab} = \frac{\mathbf{u}^+ - \mathbf{u}^-}{4}
ab=u+−u−4\mathbf{ab} = \frac{\mathbf{u}^+ - \mathbf{u}^-}{4}
\mathbf{u}^+ \geq (\mathbf{a} + \mathbf{b})^2
u+≥(a+b)2 \mathbf{u}^+ \geq (\mathbf{a} + \mathbf{b})^2
\mathbf{u}^- \geq (\mathbf{a} - \mathbf{b})^2
u−≥(a−b)2 \mathbf{u}^- \geq (\mathbf{a} - \mathbf{b})^2

[2] Ponton et al., "A Convex Model of Momentum Dynamics for Multi-Contact MotionGeneration", 2013

A gait matrix [3]                           describes when i contact swings in determined time-slot j                 :

Since each contact location in the plan is reached once, we enforce:

Contact and Gait Sequence

\sum_{j=1}^{N_t} \mathbf{T}_{i j} = 1 \ , \ \forall i = 1,..,N_f.
∑j=1NtTij=1 , ∀i=1,..,Nf.\sum_{j=1}^{N_t} \mathbf{T}_{i j} = 1 \ , \ \forall i = 1,..,N_f.
(\mathbf{T}_{ij} = 1)
(Tij=1)(\mathbf{T}_{ij} = 1)
N_f
Nf N_f
N_t
Nt N_t
\mathbf{T} \in \{0,1\}^{N_f\times N_t}
T∈{0,1}Nf×Nt\mathbf{T} \in \{0,1\}^{N_f\times N_t}
  • Number of contacts
  • Number of time-slots

[3] Aceituno-Cabezas et al., “A Mixed-Integer Convex Optimization Framework for Robust Multilegged

Robot Locomotion Planning over Challenging Terrain”, 2017

Once is assigned the contact to swing, we optimize the contact locations, represented as 4D vectors:

and then, we assign this contact to one of the       by using a binary matrix                                     :

Contact and Gait Sequence

\sum_{\mathrm{r} = 1}^{N_r}{\mathbf{H}_{i \mathrm{r}}} = 1, \mathbf{H}_{i \mathrm{r}} \Rightarrow A_\mathrm{r} \mathbf{f}_i \leq b_r,
∑r=1NrHir=1,Hir⇒Arfi≤br,\sum_{\mathrm{r} = 1}^{N_r}{\mathbf{H}_{i \mathrm{r}}} = 1, \mathbf{H}_{i \mathrm{r}} \Rightarrow A_\mathrm{r} \mathbf{f}_i \leq b_r,
\mathbf{H} \in \{0,1\}^{N_f\times N_r}
H∈{0,1}Nf×Nr\mathbf{H} \in \{0,1\}^{N_f\times N_r}
\Rightarrow
⇒\Rightarrow
N_r
Nr N_r
\mathbf{f} = (f_x, f_y, f_z, \theta),
f=(fx,fy,fz,θ),\mathbf{f} = (f_x, f_y, f_z, \theta),

where the        (implies) operator is represented with big-M formulation.

[3] Aceituno-Cabezas et al., “A Mixed-Integer Convex Optimization Framework for Robust Multilegged

Robot Locomotion Planning over Challenging Terrain”, 2017

We approximate the kinematic limits as a bounding box with respect to the CoM. Algebraically:

where

  • CoM position at i transition
  • Diagonal of the bounding box
  • Distance from the trunk of leg

Contact and Gait Sequence

\mathbf{r}_{T(i)}
rT(i)\mathbf{r}_{T(i)}
L_i
LiL_i
d_{\lim}
dlim d_{\lim}
\left | \mathbf{f}_{i} - \left[ \mathbf{r}_{T(i)} + L_i \begin{pmatrix} \cos(\theta_i + \phi_i) \\ \sin(\theta_i + \phi_i) \end{pmatrix} \right] \right | \leq d_{lim}
∣fi−[rT(i)+Li(cos(θi+ϕi)sin(θi+ϕi))]∣≤dlim\left | \mathbf{f}_{i} - \left[ \mathbf{r}_{T(i)} + L_i \begin{pmatrix} \cos(\theta_i + \phi_i) \\ \sin(\theta_i + \phi_i) \end{pmatrix} \right] \right | \leq d_{lim}

and the trigonometric functions are decomposed in piecewise linear
functions [4].

[4] Deits and Tedrake, “Footstep Planning on Uneven Terrain with Mixed-Integer Convex Optimization”, 2014

We define            as swing reference trajectory, connected to adjacent
contacts, where:

  • j indicates the time-slot, and
  •                          all the knots per time-slot

Contact and Gait Sequence

t \in [1, \cdots, N_k]
t∈[1,⋯,Nk]t \in [1, \cdots, N_k]
\mathbf{T}_{ij} \Rightarrow \mathbf{p}_{l(i)\gamma(j,N_k)} = \mathbf{f}_i,
Tij⇒pl(i)γ(j,Nk)=fi,\mathbf{T}_{ij} \Rightarrow \mathbf{p}_{l(i)\gamma(j,N_k)} = \mathbf{f}_i,
\mathbf{f}_i
fi \mathbf{f}_i

This constraint enforces that the leg reaches the contact position      at
the end of the j slot

\gamma(j,t)
γ(j,t)\gamma(j,t)

with:

The leg number for the i contact

l(i)
l(i)l(i)

For stance condition, we constraint the leg to remain stationary as:

  •          are the contact indexes assigned to the l leg

Contact and Gait Sequence

\forall t \in [2,\cdots,N_k],
∀t∈[2,⋯,Nk],\forall t \in [2,\cdots,N_k],
\sum_{i \in C(l)}^{ } \mathbf{T}_{i j} = 0 \Rightarrow \mathbf{p}_{l \gamma(j,t)} = \mathbf{p}_{l \gamma(j,1)}
∑i∈C(l)Tij=0⇒plγ(j,t)=plγ(j,1)\sum_{i \in C(l)}^{ } \mathbf{T}_{i j} = 0 \Rightarrow \mathbf{p}_{l \gamma(j,t)} = \mathbf{p}_{l \gamma(j,1)}

where

C(l)
C(l)C(l)

If a swing is activated on l leg, there is no contact force:

Contact and Gait Sequence

NC(j)
NC(j)NC(j)
\sum_{i \in C(l)}\mathbf{T}_{i j} = 1 \Rightarrow \mathbf{\lambda}_{l \gamma(j,t)} = 0 \ , \ \forall t \in NC(j),
∑i∈C(l)Tij=1⇒λlγ(j,t)=0 , ∀t∈NC(j),\sum_{i \in C(l)}\mathbf{T}_{i j} = 1 \Rightarrow \mathbf{\lambda}_{l \gamma(j,t)} = 0 \ , \ \forall t \in NC(j),

where              is the set of knots in the j slot used for the swing.

This can be see as complementary constraint [5,6]

[5] Stewart and Trinkle, “An implicit time-stepping scheme for rigid body dynamics with Coulomb friction”, 2000

[6] Mastalli et al., “Hierarchical Planning of Dynamic Movements without Scheduled Contact Sequences”, 2016

We add robustness to this condition because we maximize a lower bound of the distance of the force vector to each cone edge        as:

Contact and Gait Sequence

\rho_e
ρe\rho_e
(\alpha)
(α)(\alpha)

where      are positive multipliers on each cone edge

We can ensure stability on arbitrary terrain with non-coplanar contacts by incorporating linearized friction cone constraints:

 

\mathbf{\lambda}_{l j} \in \mathbb{FC}_\mathrm{r} \Rightarrow \mathbf{\lambda}_{l j} = \sum_{e = 1}^{N_e} \rho_e \mathbf{v}_{\mathrm{r}_e} \ , \ \rho_1, \cdots, \rho_{N_e} > 0,
λlj∈FCr⇒λlj=∑e=1Neρevre , ρ1,⋯,ρNe>0,\mathbf{\lambda}_{l j} \in \mathbb{FC}_\mathrm{r} \Rightarrow \mathbf{\lambda}_{l j} = \sum_{e = 1}^{N_e} \rho_e \mathbf{v}_{\mathrm{r}_e} \ , \ \rho_1, \cdots, \rho_{N_e} > 0,
\alpha = \text{arg}\max_{\bar{\alpha}}\text{ s.t }\mathbf{\lambda}_{l j} - \bar{\alpha} \hat{\mathbf{n}}_\mathrm{r} \in \mathbb{FC}_\mathrm{r}.
α=argmaxα¯ s.t λlj−α¯n^r∈FCr.\alpha = \text{arg}\max_{\bar{\alpha}}\text{ s.t }\mathbf{\lambda}_{l j} - \bar{\alpha} \hat{\mathbf{n}}_\mathrm{r} \in \mathbb{FC}_\mathrm{r}.

Contact and Gait Sequence

We approximate joint torque limits by using a nominal value of the Jacobian       :

\mathbf{J}_{l,j}^*
Jl,j∗\mathbf{J}_{l,j}^*
{\mathbf{J}^*}_{l,j}^{T} \mathbf{\lambda}_{l,j} \leq \mathbf{\tau}_{max},
J∗l,jTλl,j≤τmax,{\mathbf{J}^*}_{l,j}^{T} \mathbf{\lambda}_{l,j} \leq \mathbf{\tau}_{max},

HyQ Crossing Terrain with Various Slopes

[7] Mastalli et al., “Motion planning for quadrupedal locomotion: coupled planning, terrain mapping and whole-body control”, 2018 (unpublished)

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