Human-Exoskeleton Kinematics & Stiffness Models

• Human–exoskeleton kinematics and stiffness models are mathematical frameworks that describe motion compatibility, interaction forces, and compliance between the human body and robotic exoskeletons.
• These models range from reduced-order single-joint formulations to high-dimensional multibody systems, enabling accurate estimation of hysteretic damping and interface elasticity.
• They support adaptive control synthesis and stability analysis by integrating multimodal sensing, predictive simulation, and phase-dependent stiffness adjustments for improved safety and transparency.

Human–exoskeleton kinematics and stiffness modeling denotes the family of mathematical descriptions used to represent motion compatibility, interaction torques, interface compliance, and control-relevant impedance in coupled human–robot systems. In the surveyed literature, these models range from single-degree-of-freedom elbow and wrist formulations to lower-limb multibody systems, soft tendon-driven ankle devices, continuum spine exoskeletons, and therapist-mediated rehabilitation exoskeletons. Across these settings, the central problem is consistent: relate human motion, exoskeleton motion, and interaction forces in a form suitable for identification, controller synthesis, transparency analysis, and safety assessment (He et al., 2019, Jin et al., 2023, Bezzini et al., 2024).

1. Kinematic representations and coupling assumptions

A large portion of the literature begins with reduced-order joint-space models. In single-joint upper-limb systems, the human limb and exoskeleton are often treated as rigidly aligned about one rotational axis. For elbow amplification exoskeletons, the usual assumption is $q_h(t) \approx q_e(t)$, with series elastic actuation represented by the spring deflection $q_s(t)=q_m(t)-q_e(t)$. The same simplification appears in the 1-DOF elbow-joint performance augmentation exoskeleton, where the exoskeleton and forearm form a rigid linkage about the elbow and the torque $\tau_s$ is transmitted through a rigid coupling at the elbow angle $\theta_e$ (He et al., 2018, He et al., 2019, He et al., 2020).

Lower-limb models span several scales. At the compact end, knee assistance on sand is modeled in the sagittal plane with $q=[\theta_t,\theta_k]^T$, a 2-DOF planar linkage driven by human torque, exoskeleton torque, and ground-reaction-force-induced external torques. At higher dimensionality, a lower-limb rehabilitation device is represented as a 7-DOF anthropomorphic chain using modified Denavit–Hartenberg coordinates, while a coupled neuromusculoskeletal–exoskeletal gait simulator uses 11 generalized coordinates for trunk, bilateral hip, knee, ankle, and bilateral exoskeleton thigh segments. More expansive transparency studies model the human lower limbs with 18 DoFs and the exoskeleton plus harnesses with 42 DoFs, so that interface kinematics can be resolved at thigh, shank, foot, and pelvis contact sites (Zhu et al., 2024, Hasan, 2024, Jin et al., 2023, Bezzini et al., 2024).

Not all systems are well represented by a small number of revolute joints. A tendon-driven wrist mechanism treats wrist abduction–adduction as a single revolute DoF with aligned human and exoskeleton axes, but a spine-inspired soft exoskeleton uses a discretized continuum model in which $n$ rigid discs are connected by ball-and-socket joints, yielding $3n$ configuration variables and an overall transform

$T_E = T_1T_2\cdots T_n\,\mathrm{Tran}(e).$

That formulation is specifically motivated by the anatomical fact that the spine is inappropriate to simplify as a single degree-of-freedom joint (Khan et al., 21 Apr 2026, Yang et al., 2019).

These kinematic choices are not merely bookkeeping. They determine whether interaction is represented as rigid congruence, compliant relative motion, or distributed curvature, and they therefore constrain which stiffness model is mechanically meaningful.

2. Stiffness, impedance, and compliance formulations

The canonical definition of dynamic stiffness in this literature is the frequency-domain torque–angle ratio

$K^*(\omega)=\frac{T_h(\omega)}{\Theta(\omega)}.$

For a classical joint model with inertia $J$, viscous damping $q_s(t)=q_m(t)-q_e(t)$0, and real stiffness $q_s(t)=q_m(t)-q_e(t)$1,

$q_s(t)=q_m(t)-q_e(t)$2

This form underlies many exoskeleton analyses because it directly determines resonance, damping ratio, and closed-loop phase margin (He et al., 2019).

A major development is the replacement of purely real stiffness by complex stiffness,

$q_s(t)=q_m(t)-q_e(t)$3

or, equivalently in the bond-graph notation of the elbow studies, $q_s(t)=q_m(t)-q_e(t)$4. In that model, the human impedance becomes

$q_s(t)=q_m(t)-q_e(t)$5

This representation was introduced to explain two observations that the classical model fails to capture: a non-zero low-frequency phase shift in the torque–angle frequency response, and a near constant damping ratio as stiffness and inertia vary. Experiments on elbow exoskeletons showed that the hysteretic damping term improves modeling accuracy using a statistical $q_s(t)=q_m(t)-q_e(t)$6-test, and that this improvement is more statistically significant than the inclusion of classical viscous damping. The same line of work further reported an approximately linear relation between the hysteretic damping and the real stiffness, $q_s(t)=q_m(t)-q_e(t)$7, enabling reduction to a 1-parameter stiffness model (He et al., 2019, He et al., 2020).

Across the broader literature, stiffness appears in several mechanically distinct ways.

Model family	Representative relation	Context
Joint impedance	$q_s(t)=q_m(t)-q_e(t)$8	Single-joint human dynamics
Complex stiffness	$q_s(t)=q_m(t)-q_e(t)$9	Hysteretic elbow behavior
Interface or actuator elasticity	$\tau_s$0, $\tau_s$1, $\tau_s$2	Harnesses, soft suits, clock springs

In gait and interface models, stiffness is often state-dependent or phase-dependent rather than frequency-domain. Knee assistance on sand uses a piecewise linear quasi-stiffness estimate,

$\tau_s$3

with a sigmoid gate switching between stance and swing. Human–exoskeleton coupling in neuromusculoskeletal simulation and harness modeling is frequently represented by Kelvin–Voigt elements, such as

$\tau_s$4

for rotational thigh interfaces, or

$\tau_s$5

for full 6D interaction wrenches. Soft ankle exosuits use an equivalent series stiffness

$\tau_s$6

while tendon- or spring-assisted wrist devices use

$\tau_s$7

These formulations show that “stiffness” in human–exoskeleton systems may refer to joint impedance, interface compliance, suit deformation, or actuator elasticity, depending on the level of abstraction (Zhu et al., 2024, Jin et al., 2023, Bezzini et al., 2024, Tan et al., 13 Aug 2025, Khan et al., 21 Apr 2026).

3. Identification, sensing, and estimation of interaction mechanics

One identification route is direct frequency-response measurement. In elbow amplification exoskeletons, the exoskeleton applies controlled torque excitation, measures elbow angle and interaction torque, computes $\tau_s$8, and fits competing models. That workflow was used to compare real-stiffness, complex-stiffness, and combined models, and it is the basis for the conclusion that hysteretic damping is dominant in those datasets (He et al., 2019).

A second route is time-domain regression and sensor-driven learning. For elbow impedance estimation, one study used the linear relation

$\tau_s$9

as a sliding-window regression model to generate reference stiffness labels, then trained a random forest regressor on processed sEMG, stretch-sensor signals, and exoskeleton kinematics. The reported stiffness range was approximately $\theta_e$0 to $\theta_e$1, with overall correlation $\theta_e$2, maximum error $\theta_e$3, and error variance $\theta_e$4. Removing the stretch sensors reduced correlation to $\theta_e$5 and increased both maximum error and variance, indicating that local surface deformation contributed information beyond sEMG alone (Huang et al., 2019).

A third route is multimodal estimation of joint moments and physiological state. A soft leg sleeve integrating IMUs, textile sEMG, and strain sensors was used to estimate ankle joint moments with leave-one-subject-out RMSE $\theta_e$6, classify metabolic trends with $\theta_e$7 accuracy, and detect injury risk within $\theta_e$8 with recall $\theta_e$9. The same study interprets IMUs as skeletal-layer sensing, sEMG as muscular-layer sensing, and strain as cutaneous-layer sensing, thereby supplying kinematic, effort-related, and interface-related observables for impedance-aware control (Tang et al., 16 Aug 2025).

Broader system identification also appears in terrain-aware lower-limb control. Knee assistance on sand used an IMU-based dual-path Bi-LSTM plus MLP to estimate vertical and longitudinal ground reaction forces in real time, with normalized RMSE $q=[\theta_t,\theta_k]^T$0 and $q=[\theta_t,\theta_k]^T$1 on solid ground for $q=[\theta_t,\theta_k]^T$2 and $q=[\theta_t,\theta_k]^T$3, and $q=[\theta_t,\theta_k]^T$4 and $q=[\theta_t,\theta_k]^T$5 on sand. In rehabilitation, therapist–patient dyadic exoskeleton data were embedded into a shared latent manifold via a variational autoencoder and mapped into probabilistic force fields with a Gaussian mixture model, providing a stride-space representation of corrective interaction behavior (Zhu et al., 2024, Snyder et al., 2 Mar 2026).

Taken together, these methods indicate that stiffness estimation is no longer limited to direct perturbation tests. It can be inferred from multimodal sensing, latent kinematic structure, or simulation-generated labels, provided the mechanical assumptions are explicit.

4. Control synthesis, amplification, and stability

In single-joint amplification exoskeletons, stiffness models enter control through the amplification error and loop-shaping problem. One canonical formulation defines

$q=[\theta_t,\theta_k]^T$6

with $q=[\theta_t,\theta_k]^T$7 and $q=[\theta_t,\theta_k]^T$8. Because the plant $q=[\theta_t,\theta_k]^T$9 varies with both human impedance and load impedance, the compensator must remain robust across soft versus clenched arms and light versus heavy loads. Experiments showed that a slightly aggressive controller becomes borderline stable for soft human musculoskeletal behavior and a heavy load, which directly links controller margin to the assumed stiffness model (He et al., 2018).

Compliance shaping makes this dependence explicit. With the actuator law

$n$0

the exoskeleton shapes the human-side compliance while leaving the environment-side compliance $n$1 unchanged. Online stiffness estimation can then be used to place the zeros and poles of $n$2 relative to the human–exo natural frequency

$n$3

The reported result is improved bandwidth and amplification while remaining robustly stable, but an intentional mis-setting of the stiffness estimate can destabilize the system until actual co-contraction rises to match the assumed stiffness (Huang et al., 2019).

The complex-stiffness elbow literature pushes this further by designing controllers around constant-phase human dynamics. Once $n$4 is accepted, the human joint behaves like a fractional-order or hysteretic element, and fractional-order control becomes a natural match. Those studies used customizable fractional-order control to exploit hysteretic damping and improve strength amplification bandwidth while maintaining stability under muscle co-contraction variation (He et al., 2019, He et al., 2020).

Lower-limb control uses different but related formulations. On sand, knee assistance is generated by estimating human torque from phase-dependent quasi-stiffness and setting desired assistive torque as

$n$5

A stiffness-based MPC then enforces gait tracking, torque smoothness, and state constraints; the reported experiments showed muscle activation reductions of $n$6 and metabolic reduction of $n$7 on sand (Zhu et al., 2024). For multitask elbow exoskeleton control, a simulation-trained variable-impedance policy outputs both reference trajectories and stiffness gains while satisfying the Lyapunov-derived scalar bound

$n$8

which constrains stiffness increases to guarantee asymptotic stability of the impedance error dynamics (Ma et al., 4 Jun 2026).

A different use of the model appears in soft ankle assistance for non-steady locomotion. There, a dual-Gaussian desired force profile $n$9 is converted into a feedforward cable-velocity command through the coupled kinematics–stiffness relation

$3n$0

Because the independent variable is shank angle rather than time, the controller remains coordinated during phase perturbations, and the assistance profile retains high correlation with biological ankle moment patterns across walking, running, and stair negotiation (Tan et al., 13 Aug 2025).

5. Transparency, safety, and morphology-dependent interaction

Stiffness modeling is also a design tool. In lower-limb transparency studies, the exoskeleton and its contact mechanisms are simulated with 6D spring–damper interface elements, and transparency is quantified by the quadratic wrench cost

$3n$1

Interface stiffness and damping are optimization variables, while nonlinear distance constraints ensure the exoskeleton remains engaged with the wearer. In that framework, the virtual harness configuration $3n$2 preserved tracking quality comparable to a more rigid $3n$3 configuration while reducing overall interaction wrenches, illustrating how passive self-alignment and impedance distribution can improve transparency (Bezzini et al., 2024).

Safety-critical collision behavior depends strongly on joint compliance. In a lower-limb test bench with a MACCEPA variable-stiffness knee, reducing knee stiffness lowered the peak pelvis torque transmitted during swing-phase collision from $3n$4 to $3n$5, and reduced the mechanical impulse by a factor of three. The same experiments showed that compliant knees tend to redirect motion upward over the obstacle rather than reflecting impact backward, which changes both peak force transmission and post-impact kinematics (Schrade et al., 2019).

Morphology-specific load models provide a different form of stiffness relevance. A continuum soft spine exoskeleton used a virtual impedance controller to generate assistive force and compensate Bowden-cable hysteresis, while an analytical biomechanics model and a musculoskeletal simulation predicted reduced spinal loads. Under a $3n$6 assistive force during stoop lifting, the reported reductions were $3n$7 in disc compression force, $3n$8 in disc shear force, and $3n$9 in average erector spinae muscle force (Yang et al., 2019).

Range-of-motion optimization is another manifestation of kinematic–stiffness coupling. In a four-bar knee exoskeleton for sit-to-stand, non-linear kinematic optimization of link lengths increased maximum ROM to $T_E = T_1T_2\cdots T_n\,\mathrm{Tran}(e).$0, which was reported as $T_E = T_1T_2\cdots T_n\,\mathrm{Tran}(e).$1 greater than the previous design for the same actuator stroke. The subsequent mannequin, cardboard dummy, and video-tracking studies were specifically intended to examine knee-joint misalignment between human and device trajectories (Gautam et al., 2024).

6. Limitations, contested assumptions, and emerging directions

A recurrent point of disagreement concerns how human damping should be represented. Classical real-stiffness plus viscous-damping models predict zero low-frequency phase, whereas elbow experiments reported non-zero low-frequency phase and nearly constant damping ratio across stiffness and inertia changes; those observations motivated the complex-stiffness and hysteretic-damping formulations. The available evidence in the elbow literature therefore favors complex stiffness over purely viscous damping for that application, but this does not imply that the same model is universally sufficient for all joints, tasks, or attachment conditions (He et al., 2019, He et al., 2020).

Another persistent limitation is the prevalence of rigid-alignment assumptions. Many upper-limb and single-joint models set $T_E = T_1T_2\cdots T_n\,\mathrm{Tran}(e).$2 or assume perfect axis alignment, while wrist, ankle-suit, and lower-limb harness studies explicitly acknowledge interface compliance, suit migration, soft-tissue deformation, or latent misalignment. This suggests that rigid coupling is often a useful first approximation for control design, but not a general description of physical interaction (He et al., 2018, Khan et al., 21 Apr 2026, Tan et al., 13 Aug 2025, Bezzini et al., 2024).

The same pattern holds for dimensionality and linearity. One literature branch uses single-DOF, linear, or quasi-linear impedance models because they are identifiable and controller-friendly; another uses matrix-valued multibody dynamics, 6D interface wrenches, neuromusculoskeletal feedback loops, or simulation-trained variable-impedance policies because those are better suited to gait, multi-contact interaction, or task diversity. A plausible implication is that future human–exoskeleton stiffness models will increasingly combine low-dimensional control-oriented representations with higher-dimensional sensing and simulation priors, rather than replacing one class with the other outright (Jin et al., 2023, Ma et al., 4 Jun 2026, Tang et al., 16 Aug 2025).

Across the surveyed work, the field is moving toward three convergent directions. First, stiffness is being treated as an adaptive variable rather than a fixed constant, whether through random-forest stiffness estimation, stride-wise profile updates, or policy-predicted impedance gains. Second, interaction is being modeled at multiple layers simultaneously—joint, interface, and physiological—using IMUs, sEMG, strain, contact forces, and latent kinematic manifolds. Third, stability guarantees are becoming more explicit, ranging from frequency-domain phase-margin arguments to Lyapunov bounds on stiffness variation. In that sense, the modern human–exoskeleton kinematics and stiffness model is no longer just a geometric mechanism model or a mass–spring–damper surrogate; it is an integrated description of coupled motion, compliance, sensing, and control.