Series Summary: Convex MPC for Quadruped Locomotion

The Control Problem

A quadruped robot must maintain stability while executing velocity commands across uncertain terrain. The central challenge is that ground contacts are discrete events — feet turn on and off — creating nonlinear mode switches that traditional inverse kinematics cannot handle.

The MPC Solution

We formulate a finite-horizon optimal control problem that centralizes all decisions around ground-reaction forces:

Given: Current body state $\mathbf{x}_0$ (position, orientation, velocities) and velocity command $\dot{\mathbf{p}}^{\text{cmd}}$

Optimize: Forces $\{\mathbf{f}_{i,k}\}$ at all four feet over a 400 ms horizon (16 × 25 ms steps)

Minimize: State tracking error + force effort $J = \sum_{k=0}^{15} \left( \|\mathbf{x}_k - \mathbf{x}^{\text{ref}}_k\|_Q^2 + \|\mathbf{u}_k\|_R^2 \right)$

Subject to:

• Single-rigid-body dynamics (Part 1–2): $\mathbf{x}_{k+1} = \Phi \mathbf{x}_k + \Gamma \mathbf{u}_k + \mathbf{c}$
• Friction cones at feet: $\|\mathbf{f}_{i,xy}\| \leq \mu f_{i,z}$ (Part 3)
• Fixed-timing gait schedule: $\mathbf{f}_{i,k} = 0$ during swing (Part 4)

Key Innovations

1. Single Rigid Body Model (Part 1–2)

• Simplifies quadruped as 13-state system (position, orientation, velocities)
• Assumes leg dynamics are fast → forces transmit instantaneously to body
• Valid for typical trotting speeds (|v| under 1 m/s)
• Reduces control complexity from multi-body to single-body

2. Linearization and Discretization (Part 2)

• Linearizes nonlinear dynamics around reference trajectory
• Discretizes at 40 Hz (25 ms intervals) with zero-order hold
• Enables convex optimization (linear constraints + quadratic cost)
• Linearization errors under 10% for trotting

3. Condensed QP (Part 3)

• Eliminates state variables → only 192 force variables
• Matrices $H$, $c$ implicitly encode full dynamics
• Solves in under 5 ms via qpOASES (warm-started)
• Convex problem guarantees global optimum

4. Fixed-Timing Gait (Part 4)

• Diagonal trot: alternates front-left/back-right with front-right/back-left
• Predicts contact schedule in advance → convex force constraints
• No foot sensors needed; robust to small timing errors
• Tradeoff: cannot adapt to rough terrain

5. Two-Level Control (Part 4)

• Upper level (MPC, 40 Hz): Plans ground-reaction forces for balance
• Lower level (PD, 1 kHz): Tracks swing-leg trajectories independently
• Clean separation: MPC handles "what forces?" + "when?" → swing control handles "where foot goes?"

Performance

Tracking accuracy (60 s runs):

• Linear velocity: ±5% of command
• Yaw rate: ±10° over 20+ meter walk
• Lateral drift: under 1 mm per meter traveled
• Body height: Reference ± 3 cm

Control timing:

• MPC solve: 2–4 ms
• Total cycle: ~5–7 ms (25 ms budget)
• Headroom: ~8 ms @ 40 Hz planning rate

Embodiment:

• Tested on ANYmal C (28 kg quadruped with 12 DOF)
• Same code runs on MIT Cheetah 3 and other quadrupeds
• Compiles to WebAssembly for browser visualization

Why This Approach Works

1. Problem centering: Optimizing forces directly (not joint angles) aligns with physics
2. Convex formulation: Friction cones are convex → no local minima, guaranteed convergence
3. Real-time feasible: 192-variable QP solves in 5 ms with warm-starting
4. Modular: Single-body model reusable across robot variants
5. Robust: Fixed gait eliminates sensor dependencies for feet contact

Limitations and Extensions

Current simplifications:

• Single rigid body (ignores leg inertia)
• No contact detection (fixed timing can mismatch terrain)
• Proportional cost only (small steady-state errors)
• No integral action or disturbance rejection
• Assumes flat or gently sloped terrain

Possible extensions:

• Adaptive gait switching (bound, pace, gallop)
• Machine learning for terrain adaptation
• Contact state estimation with foot sensors
• Whole-body control (arms + legs)
• Learned dynamics models or cost functions

Implementation Highlights

Code structure:

• ConvexMPC/ — Simulator-independent controller (Eigen + qpOASES)
• simulation/ — MuJoCo integration + headless CLI
• wasm/ — Emscripten bindings for browser
• web/ — React + Three.js interactive viewer

Deployment options:

• Native: 100× real-time simulation for testing
• WebAssembly: Interactive browser demo
• Hardware: Real-time capable (deterministic, predictable)

Recommended Reading Order

1. Part 1 — Physics foundation: why single rigid body, contact constraints
2. Part 2 — Linearized dynamics: why we can solve a convex QP
3. Part 3 — The QP problem: cost, constraints, condensation trick
4. Part 4 — Gait and swing: how everything fits together in time
5. Part 5 — Implementation: solver tricks, tuning, real-time performance
6. This summary — Big picture: why MPC, what works, what doesn't

Source and References

Repository: github.com/CMaybe/Convex-MPC

Key paper:

J. Di Carlo, P. M. Wensing, B. Katz, G. Bledt, and S. Kim, "Dynamic Locomotion in the MIT Cheetah 3 Through Convex Model-Predictive Control," 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 7440-7447. doi: 10.1109/IROS.2018.8594448

Dependencies:

• Eigen 3.4+ (linear algebra)
• qpOASES (quadratic programming)
• MuJoCo 3.1+ (physics simulation)
• Emscripten (WebAssembly compilation)

This series demonstrates how modern optimal control—specifically convex model predictive control—enables real-time, provably-stable locomotion for complex legged robots. The approach trades model sophistication for computational efficiency and robustness, enabling dynamic walking at 40 Hz on commodity hardware.