Force-Compliance MPC

• Force-Compliance MPC is a control approach that integrates force estimation and compliance modeling into predictive planning for dynamic physical interactions.
• It employs techniques such as wrench estimation, stiffness adaptation, and flexible-joint dynamics to incorporate force feedback directly into motion predictions.
• The method unifies safety, dynamic response, and compliance constraints in a single optimization framework to enhance robotic performance in varying contact scenarios.

Force-Compliance Model Predictive Control (FC-MPC) designates receding-horizon control schemes that couple motion planning with force-responsive or compliance-aware behavior during physical interaction. In the narrowest and explicit sense, FC-MPC estimates externally applied forces and moments, converts them into a force-compliance motion target, and optimizes future robot motion under dynamic and safety constraints (Fan et al., 5 Aug 2025). Closely related work extends the same design problem in several directions: compliance can enter the predictive model through deformation-dependent centroidal inertia in legged locomotion (Ye et al., 28 Apr 2025), through stiffness adaptation in variable impedance control (Anand et al., 2022), through flexible-joint torque dynamics and actuator constraints (Iskandar et al., 2022), or through learned force/moment output models with stochastic safety guarantees (Matschek et al., 2023). Force-centric rigid-contact MPC, although not compliance-aware in the strict sense, remains an important precursor because it treats contact reaction forces as the principal predictive control variable rather than a by-product of body-trajectory tracking (Kim et al., 2019).

1. Scope and conceptual boundaries

A useful distinction suggested by these works is between explicit FC-MPC, compliance-aware force MPC, and adjacent force/compliance MPC formulations. In explicit FC-MPC, interaction wrench estimates directly bias the MPC objective toward compliant motion while safety constraints remain part of the same optimization problem. In compliance-aware force MPC, the optimized variable is still contact force, but the predictive dynamics are modified so that embodied compliance changes the force allocation itself. In adjacent formulations, MPC regulates impedance parameters, flexible-joint torque dynamics, or force/motion outputs under uncertainty, without necessarily optimizing a force-compliance law in the narrow sense.

Approach	Representative formulation	FC-MPC status
Direct force-compliance MPC	user wrench estimation, force-compliance velocity, MPC, and Robot-User CBFs	explicit FC-MPC (Fan et al., 5 Aug 2025)
Compliance-aware centroidal MPC	CCPDI-enabled GRF optimization for embodied compliance	explicit compliance in predictive dynamics (Ye et al., 28 Apr 2025)
Variable-impedance MPC	stiffness adaptation in a low-level Cartesian VIC	indirect force/compliance shaping (Anand et al., 2022)
Flexible-joint MPC	SP-based MPC over slow, fast, or full joint dynamics	supporting compliant-actuation layer (Iskandar et al., 2022)
Safe force-and-motion MPC	GP-based force/output model with chance constraints	adjacent force MPC with safety guarantees (Matschek et al., 2023)
Force-centric rigid-contact MPC	reaction-force MPC with WBIC realization	precursor on the force side (Kim et al., 2019)

Within this spectrum, the central technical question is not merely whether force appears in the formulation, but where compliance is represented. The cited work places compliance in at least four different locations: in a force-to-velocity mapping inside the objective, in the predictive inertia model, in flexible-joint internal dynamics, or in the impedance parameters of a low-level controller. This suggests that FC-MPC is best treated as a family of architectures rather than a single canonical optimization template.

2. Canonical explicit formulation

The most direct FC-MPC formulation in the cited material appears in the hexapod guide-robot framework, where full-body dynamics are used to estimate a user-applied wrench and a reduced planar model is used for receding-horizon planning. The force-estimation layer starts from

$$M(q)\ddot q + C(q,\dot q) + G(q) = \tau + J^\top \Lambda,$$

with the generalized external force partitioned into base wrench and foot-contact terms. A Recursive Least Squares update produces an estimate of the base wrench, and the planar channels $(\hat F_x,\hat F_y,\hat M_z)$ are accumulated into an impulse-like quantity

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$

which is mapped to a force-compliance velocity

$$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$

The reduced MPC state is $x_k=[x_k\; y_k\; \theta_k]^\top$, the control is $u_k=[v_{xk}\; v_{yk}\; \omega_k]^\top$, and the discrete kinematics are

$$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k\ y_k\ \theta_k \end{bmatrix} + \begin{bmatrix} \cos\theta_k & -\sin\theta_k & 0\ \sin\theta_k & \cos\theta_k & 0\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} v_{xk}\ v_{yk}\ \omega_k \end{bmatrix}\Delta t.$$

The FC-MPC objective is written as

$$\min \; p(x_Z) +\sum_{k=0}^{Z-1}\big(q(x_k)+v(u_k)\big) +\sum_{k=0}^{Z-1}\sum_{i=1}^{2}K_i\delta_{i,k},$$

with stage and terminal tracking terms for navigation, velocity limits $u_k\in U$, and a force-compliance term

$$v(u_k)=\|u_k-\mu^k V_{\text{fc},t}\|_R+\|u_k-u_{k-1}\|_S.$$

The geometric decay $(\hat F_x,\hat F_y,\hat M_z)$0 acts as a virtual damping factor, so the future compliant velocity target decays across the horizon rather than being perpetuated indefinitely. When the applied wrench does not exceed a threshold, $(\hat F_x,\hat F_y,\hat M_z)$1, and the controller degrades into a standard MPC. This is an explicit example of force compliance entering the optimizer through a force-induced motion target rather than through direct force tracking (Fan et al., 5 Aug 2025).

A notable structural feature is that safety is not appended as a separate filter. Robot-User Control Barrier Functions are embedded as soft constraints of the form

$(\hat F_x,\hat F_y,\hat M_z)$2

and the associated slack penalties are weighted so that user safety can be prioritized over robot safety when the safe sets conflict. This places force responsiveness, autonomous navigation, and safety resolution inside a single optimization problem (Fan et al., 5 Aug 2025).

3. Where compliance enters the predictive model

One FC-MPC route makes compliance explicit in the predictive dynamics themselves. In deformable legged locomotion, a compliant spine changes the robot’s mass distribution, centroid location, and inertia tensor over the horizon. The CCPDI formulation approximates a deformable body by rigid sub-bodies whose relative motion represents deformation, recursively composes a predictive deformed composite inertia, and injects it into an otherwise standard centroidal MPC. The key insertion is

$(\hat F_x,\hat F_y,\hat M_z)$3

so compliance changes the matrices in the centroidal prediction model without changing the standard convex MPC structure. The control input remains the stacked ground reaction force vector

$(\hat F_x,\hat F_y,\hat M_z)$4

The reported effect is that CCPDI-enabled MPC distributes the ground reactive forces closer to the heuristics for body balance and stabilizes the compliant robot under the same MPC configurations used for the rigid robot (Ye et al., 28 Apr 2025).

A second route places compliance in intrinsic actuator or joint elasticity. Flexible-joint robots are modeled by

$(\hat F_x,\hat F_y,\hat M_z)$5

$(\hat F_x,\hat F_y,\hat M_z)$6

or, in torque coordinates,

$(\hat F_x,\hat F_y,\hat M_z)$7

$(\hat F_x,\hat F_y,\hat M_z)$8

Singular perturbation decomposes these dynamics into slow link dynamics and fast torque dynamics, and the paper studies three linear MPC architectures: MPC-fast, MPC-slow, and MPC-full. The most FC-MPC-relevant case is MPC-fast, which explicitly regulates the elastic torque dynamics and is paired experimentally with an outer link-side position/impedance loop, making it a supporting reference for compliance-aware inner-loop MPC rather than a full task-level force-compliance formulation (Iskandar et al., 2022).

A third route places compliance in impedance parameters rather than in explicit force or deformation states. In Cartesian variable impedance control, the desired behavior is

$(\hat F_x,\hat F_y,\hat M_z)$9

with acceleration command

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$0

A high-level CEM-based MPC optimizes the future stiffness sequence $$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$1, while damping is tied to stiffness by

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$2

The predictive model is a probabilistic ensemble neural network that forecasts the next task-space state from current state, stiffness, external force, and reference. This is not direct force tracking; force enters as a measured disturbance and compliance is shaped by online impedance adaptation (Anand et al., 2022).

By contrast, the force-centric quadruped controller based on MPC and whole-body impulse control is almost entirely on the rigid-contact side. Its predictive model is the centroidal single-rigid-body dynamics

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$3

which are discretized into

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$4

The MPC plans reaction forces directly, while WBIC realizes those forces through a lower QP that softens force tracking and relaxes floating-base acceleration tracking. This architecture is highly relevant to FC-MPC because it shows why reaction-force commands can be more meaningful than exact body-trajectory commands in dynamic gaits with aerial phases, yet it does not include explicit environment stiffness, actuator elasticity, contact deformation, or force-error feedback laws inside MPC (Kim et al., 2019).

4. Learning, uncertainty, and safety-constrained force compliance

Learning-based FC-MPC-adjacent methods differ primarily in whether learning targets dynamics, outputs, or impedance behavior. In safe force-and-motion MPC, the uncertain interaction model appears only in the output equation,

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$5

where the learned correction is a Gaussian process. For a noisy force observation,

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$6

the GP posterior mean and variance are used in two separate ways: the mean enters the path-following error, and the variance tightens output constraints through chance constraints

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$7

Because uncertainty is confined to the static output map rather than the state dynamics, the stochastic MPC remains computationally efficient. The formulation directly supports force tracking, force limitation, and simultaneous motion/force objectives, but it does not impose an impedance or admittance law and does not model environment compliance as a dynamic interaction subsystem (Matschek et al., 2023).

In the learned variable-impedance approach, the model is instead a generalized Cartesian impedance model learned with a probabilistic ensemble neural network. Exploration maximizes information gain via ensemble disagreement,

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$8

and the MPC cost trades state error against stiffness magnitude,

$$L_N = \sum_{n=N-N_I+1}^{N} \gamma^{N-n} \begin{bmatrix} \hat F_{\text{base},x,n}\ \hat F_{\text{base},y,n}\ \hat M_{\text{base},z,n} \end{bmatrix} \Delta t,$$9

This produces the characteristic “soft when possible, stiff when necessary” behavior, but contact discontinuities are not explicitly represented, and no formal stability or passivity guarantee is given (Anand et al., 2022).

In direct FC-MPC for interactive guide robots, safety is resolved through Robot-User CBFs, obstacle clustering by Eight-Way Connected DBSCAN, Minimum Bounding Ellipse obstacle models, and Kalman prediction of obstacle trajectories. The weighted slack structure of the embedded CBF constraints addresses feasibility issues in complex dynamic environments and allows asymmetric prioritization of user safety over robot safety (Fan et al., 5 Aug 2025). A plausible implication is that FC-MPC systems can separate force compliance from safety only at some loss of coordination; the cited guide-robot design instead treats them as coupled optimization terms.

5. Robotic domains and reported results

The cited work spans legged locomotion, flexible-joint actuation, compliant manipulation, safe contact-sensitive manipulation, and interactive navigation. Representative reported results are summarized below.

Paper	Platform and task	Reported result
(Kim et al., 2019)	MIT Mini-Cheetah; six gaits in different environments	stable top speed $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$0; momentary observed maximum $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$1 before losing balance
(Iskandar et al., 2022)	single flexible joint with DLR C-Runner elastic element and DLR LWR III drive	chirp RMSE for MPC-fast: $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$2 rad position, $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$3 rad/s velocity
(Anand et al., 2022)	Franka Emika Panda; compliant holding, catching, pushing, drawer opening	$$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$4 additional transfer samples for task b and task c
(Matschek et al., 2023)	KUKA lightweight robot; writing-like task on flexible surface	hybrid spring+GP RMSE $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$5 N; tightened force bounds $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$6 N from $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$7 N
(Ye et al., 28 Apr 2025)	Webots 2023b quadruped with rigid or compliant spine; trot stepping	compliant robot with CCPDI-disabled MPC fails after $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$8 s; CCPDI improves average inertia prediction accuracy by about $$V_{\text{fc},N} = V_{\text{fc},0} + W^{-1}L_N,\qquad W=\mathrm{diag}(m,m,J_z).$$9
(Fan et al., 5 Aug 2025)	HexGuide hexapod guide robot in cluttered environments	robot-user soft CBF success rates $x_k=[x_k\; y_k\; \theta_k]^\top$0 in Q3 and $x_k=[x_k\; y_k\; \theta_k]^\top$1 in Q4

These results illustrate different notions of success. In force-centric legged MPC, success is high-speed dynamic locomotion with aerial phases and robustness on slippery grass and gravel (Kim et al., 2019). In flexible-joint MPC, success is oscillation damping under torque limits, where MPC-fast outperforms SP and the other MPC variants in dynamic tracking, and all MPC variants respect actuator torque limits as frequency increases (Iskandar et al., 2022). In learned impedance MPC, the main result is transferability: the same learned model is reused across tasks by changing only the MPC objective weights, with no additional samples required for transfer to the reported tasks (Anand et al., 2022).

In safe learning-supported force-and-motion MPC, the emphasis is force-model accuracy and probabilistic safety. The hybrid spring+GP model reduces force-model RMSE relative to linear and nonlinear first-principles baselines, and chance-constrained tightening prevents contact loss under disturbance in simulation while improving force control performance on hardware (Matschek et al., 2023). In compliance-aware centroidal MPC, the salient effect is not a new force objective but a better force distribution: the compliant robot stabilizes under CCPDI-enabled MPC and fails under CCPDI-disabled MPC, while the planned GRFs become more symmetric and closer to even-load-sharing heuristics (Ye et al., 28 Apr 2025). In the guide-robot FC-MPC, the principal evidence is shared-control behavior under force input together with obstacle avoidance, including full-system success-rate improvements when Robot-User soft CBFs are used (Fan et al., 5 Aug 2025).

6. Relation to adjacent control paradigms

A recurrent source of confusion is the assumption that any force-aware MPC is automatically FC-MPC. The cited work argues for a narrower interpretation. Force-centric rigid-contact MPC plans reaction forces directly and relaxes base tracking, but it does not model environment stiffness, actuator elasticity, or deformation states; it is therefore best regarded as a closely related force-based MPC method without explicit compliance modeling (Kim et al., 2019). Safe force-and-motion MPC with GP-based output models jointly tracks motion and force outputs under stochastic safety constraints, yet it does not impose a target impedance or admittance law and does not include dynamic environment compliance inside the controller (Matschek et al., 2023).

A second misconception is that compliance must be located at the contact interface. The cited work shows at least three distinct locations for compliance: embodied compliance in the robot morphology, particularly a compliant spine (Ye et al., 28 Apr 2025); intrinsic joint elasticity in lightweight robots (Iskandar et al., 2022); and virtual Cartesian compliance specified by impedance matrices $x_k=[x_k\; y_k\; \theta_k]^\top$2 (Anand et al., 2022). This suggests that FC-MPC should not be restricted to soft-contact surface models alone.

A third misconception is that force compliance always requires direct force-reference tracking. The guide-robot formulation uses estimated wrench to generate a compliant velocity target inside MPC rather than a force trajectory (Fan et al., 5 Aug 2025). The deep variable-impedance controller adapts stiffness so that force behavior is shaped indirectly through the relation $x_k=[x_k\; y_k\; \theta_k]^\top$3 (Anand et al., 2022). By contrast, the safe force-and-motion MPC directly includes force in the output vector and in chance-constrained safety sets (Matschek et al., 2023). These are materially different mechanisms, even though each belongs to the broader force/compliance predictive-control landscape.

7. Limitations and emerging directions

The principal limitations are architectural rather than merely numerical. The direct FC-MPC guide-robot paper omits exact MPC solver details, horizon values, most numerical weights, and a quantitative validation of force-estimation accuracy against ground-truth force sensors; it also notes sensitivity to perception delays, state-estimation errors, and computational load on low-power platforms (Fan et al., 5 Aug 2025). The compliance-aware centroidal MPC for embodied compliance assumes that current sub-body twists remain time-invariant over the prediction horizon, retains the small roll/pitch simplification of standard centroidal MPC, and reports only simulation results (Ye et al., 28 Apr 2025). The learned variable-impedance controller does not support non-continuous contacts, relies on CEM with limited control frequency, and provides no formal stability or robustness guarantees (Anand et al., 2022).

The supporting flexible-joint MPC paper is limited to a single-joint experimental platform, neglects friction in the core prediction models, and does not formulate explicit force objectives, contact models, or task-level force-compliance trade-offs (Iskandar et al., 2022). The safe force-and-motion MPC framework assumes that uncertainty resides only in the static output map, not in the robot dynamics, and its detailed demonstration focuses mainly on a single normal-force component (Matschek et al., 2023). The force-centric locomotion precursor remains on the rigid-contact side, with unilateral friction-pyramid constraints and no explicit compliance states in either the MPC or WBIC layers (Kim et al., 2019).

A plausible implication is that future FC-MPC systems will combine several of these ingredients rather than adopting only one. The cited works already point toward a composite architecture in which horizon-level force planning is retained from force-centric legged MPC, compliance-aware predictive models modify the force optimizer when morphology deforms, flexible-joint or impedance layers regulate compliant actuation locally, and stochastic or barrier-based safety constraints preserve safe interaction under uncertainty. In that sense, FC-MPC is less a single algorithm than a converging research program on how predictive control should represent force, compliance, and safety simultaneously.