Residual Deep Reinforcement Learning-Based Computed Torque Control for a Cable-Driven Lower-Limb Rehabilitation Robot under Disturbances and Parametric Uncertainties

Mohammad-Hossein Fakouri; Ali Keymasi-Khalaji

cs.ROAug 27, 2026

Residual Deep Reinforcement Learning-Based Computed Torque Control for a Cable-Driven Lower-Limb Rehabilitation Robot under Disturbances and Parametric Uncertainties

Authors: Mohammad-Hossein Fakouri, Ali Keymasi-Khalaji

Organizations: Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

Abstract

Accurate trajectory tracking in cable-driven lower-limb rehabilitation robots is challenging because model uncertainty, external disturbances, joint constraints, and pull-only cable actuation can degrade nominal control performance. Conventional model-based controllers provide an interpretable control structure but remain sensitive to model mismatch, whereas fully learning-based control can reduce transparency and complicate constraint-aware operation. This study proposes a residual deep reinforcement learning-enhanced computed torque control framework in which computed torque control generates the nominal command and a bounded Deep Deterministic Policy Gradient policy supplies only an additional compensating torque. The approach is evaluated in simulation under nominal, uncertain, disturbed, combined, and generalization conditions, together with trajectory-tracking, joint-limit, cable-demand, workspace-feasibility, and cable-Jacobian diagnostics. Across the evaluated conditions, the residual controller improves tracking and disturbance rejection relative to computed torque control while preserving the interpretable model-based command structure and satisfying the reported feasibility checks in the representative evaluation. Broader tests indicate that tracking improvements can persist beyond the representative case while also exposing trajectory-dependent constraint limitations. These results support bounded residual learning as a practical robustness-enhancement strategy for simulation-based rehabilitation robot control and motivate further constraint-aware and experimental validation.

Figures & tables

Literature group	Main emphasis	Role in the present study
Cable-driven rehabilitation systems	Mechanism design, low moving inertia, task-specific assistance, and human-oriented control	Defines the application domain and motivates reproducible lower-limb motion generation
Cable feasibility and CDPR control	Positive-tension feasibility, wrench distribution, force allocation, robustness, and Jacobian properties	Motivates nonnegative-tension feasibility, Jacobian-rank, minimum-singular-value, and conditioning analyses
Nonlinear rehabilitation control	Computed-torque, robust/adaptive, impedance, admittance, and assist-as-needed control	Establishes the interpretable model-based control layer and comparative baseline
DDPG and residual RL	Continuous-action learning and learned correction around an existing controller	Provides the bounded compensation mechanism used here
Present study	Computed-torque control with bounded residual DDPG for a planar 3-DOF, three-cable lower-limb model	Integrates trajectory-tracking evaluation with joint-limit, cable-tension, workspace-feasibility, and Jacobian-conditioning analyses

Table 1: Technical positioning of the present study relative to the principal literature groups.

Symbol	Meaning
$\bm{q},\dot{\bm{q}},\ddot{\bm{q}}$	Joint position, velocity, and acceleration vectors
$\bm{x},\bm{x}_{d}$	Actual and desired Cartesian end-effector positions
$\bm{p}_{i},\bm{p}_{ci}$	Joint and link-center-of-mass positions
$\bm{a}_{i},\bm{r}_{i}$	Fixed anchor and moving attachment of cable $i$
$l_{i},\bm{u}_{i}$	Cable length and anchor-to-attachment unit vector
$\bm{J}_{c}$	Implemented cable actuation Jacobian, $-\partial\bm{l}/\partial\bm{q}$

Table 2: Main symbols in the kinematic, dynamic, and control formulation.

Panel of Figure 1 — Figure 1: Planar lower-limb CDRR geometry used in the simulation model. Panel (a) defines the three-link serial mechanism and joint coordinates; panel (b) shows the fixed cable anchors, link-side attachment points, and pull directions that determine the three cable lengths.

Quantity	Value	Description
$b_{1},b_{2},b_{3}$	$0.45,0.35,0.21$ m	Calibrated nominal link lengths
$m_{1},m_{2},m_{3}$	$11.125,5.05,1.38$ kg	Link masses
$I_{cg,1},I_{cg,2},I_{cg,3}$	$0.149187,0.0522254,0.004784$ kg m 2	COM inertias
$g$	$9.81$ m s -2	Gravitational acceleration
$\bm{q}_{\min}$	$[80,2,250]^{T}$ deg	Lower joint limits
$\bm{q}_{\max}$	$[250,160,330]^{T}$ deg	Upper joint limits

Table 3: Nominal robot parameters, cable anchors, and active joint limits.

Parameter	Value
Observation/action dimensions	12 / 3
Residual torque bounds	$\pm[5,3,2]^{T}$ N m
Actor hidden layers	256 and 256 ReLU neurons
Critic feature width	256 neurons per path/stage
Agent sample time	$T_{s}=0.01$ s
Discount factor	$\gamma=0.99$

Table 4: Archived residual DDPG implementation and training settings.

Figure 2: Hybrid CTC–residual DDPG control architecture and parallel cable-demand audit. Constrained reference generation supplies the desired joint states and derivatives; CTC produces the nominal model-based torque

\bm{\tau}_{\mathrm{ctc}}

, and the trained actor supplies the bounded correction

\bm{\tau}_{\mathrm{RL}}

. Their sum

\bm{\tau}_{\mathrm{cmd}}=\bm{\tau}_{\mathrm{ctc}}+\bm{\tau}_{\mathrm{RL}}

is applied directly to the torque-driven simulated plant. In parallel,

\bm{\tau}_{\mathrm{cmd}}

and the measured joint coordinates are supplied to an algebraic cable-demand audit, whose inferred output is logged but does not drive the plant. The external disturbance

\bm{\tau}_{d}

enters the plant separately. Plant-dynamics parameter mismatch is active from 1 to 5 s, while the calibrated kinematic geometry remains fixed; disturbance is active from 5 to 9 s.

Figure 3: Residual DDPG actor–critic structure and off-policy training. The unnormalized 12-dimensional observation is mapped by the actor to three bounded residual torques. The critic estimates

Q(\bm{o},\bm{a})

from separate observation and action feature paths. Replay-buffer sampling, gradient updates, and soft target-network updates are used during training; deterministic evaluation uses the saved actor without exploration noise or online learning.

Step	Operation
1	Read $\bm{q},\dot{\bm{q}}$ and form $\bm{e},\dot{\bm{e}}$ using the constrained reference.
2	Evaluate $\bm{D}_{n},\bm{C}_{n},\bm{G}_{n}$ and compute $\bm{\tau}_{\mathrm{ctc}}$ from Eqs. ( 39 )–( 40 ).
3	Form $\bm{o}_{t}$ and evaluate the actor; add exploration only during training.
4	Scale the actor output to the bounds in Eq. ( 44 ) and form $\bm{\tau}_{\mathrm{cmd}}=\bm{\tau}_{\mathrm{ctc}}+\bm{\tau}_{\mathrm{RL}}$ .
5	Integrate the uncertain/disturbed generalized-coordinate plant and compute the reward.
6	During training, store $(\bm{o}_{t},\bm{a}_{t},r_{t},\bm{o}_{t+1})$ , sample a minibatch, update critic and actor, and softly update target networks.

Table 5: Controller execution and DDPG training/evaluation sequence.

Figure 4: Cartesian model-consistency comparison for the analytical equation-of-motion, Simscape Multibody, and MSC ADAMS implementations under the same nominal reference. Panels (a) and (b) compare the end-effector

x

- and

y

-coordinate histories, respectively.

Figure 5: Joint-angle model-consistency comparison among the analytical equation-of-motion, Simscape Multibody, and MSC ADAMS implementations. Panels (a)–(c) report $q_{1}$ , $q_{2}$ , and $q_{3}$ , respectively, under the same nominal reference.

Figure 6: Simscape-minus-equation-of-motion joint-angle differences for

q_{1}

q_{2}

, and

q_{3}

in panels (a)–(c), respectively. The difference histories localize the remaining implementation discrepancy, which is largest in the third joint channel.

Signal	Unit	Simscape–EOM	ADAMS–EOM
$q_{1}$	deg	$0.102\,/\,0.233$	$0.091\,/\,0.224$
$q_{2}$	deg	$0.409\,/\,0.984$	$0.362\,/\,0.832$
$q_{3}$	deg	$1.715\,/\,3.537$	$1.684\,/\,3.023$
$x$	mm	$3.050\,/\,7.877$	$2.959\,/\,7.812$
$y$	mm	$3.742\,/\,8.628$	$3.692\,/\,8.609$

Table 6: Archived model-consistency discrepancies relative to the analytical equation-of-motion implementation. Each entry reports RMS

/

maximum absolute discrepancy.

Check	Purpose	Interpretation
Forward/inverse kinematics	Verify Cartesian-reference and joint-coordinate consistency	Kinematic consistency of the implemented reference-generation chain
Simscape Multibody	Compare the analytical dynamics with an independent multibody formulation	Nominal simulation-to-simulation consistency under matched parameters
MSC ADAMS	Provide an additional independent multibody implementation check	Cross-platform nominal dynamic consistency
Tension post-processing	Audit the pull-only cable-demand mapping	Generalized-torque demand mapped to inferred cable tension
Workspace and $\bm{J}_{c}$ analysis	Evaluate sampled reachability, static nonnegative-tension feasibility, rank, and conditioning	Sampled, trajectory-specific cable-feasibility evidence

Table 7: Purpose and interpretive scope of the simulation checks.

Setting	Value
Simulation duration / fixed step	10 s / 0.01 s
Archived solver setting	Fixed-step ode3 under FixedStepAuto
Nominal Cartesian initial state	$[-0.90,-0.05]^{T}$ m
Primary trajectory	Eq. ( 14 )
CTC gains	$K_{p}=60I_{3}$ , $K_{d}=20I_{3}$
Uncertainty	Simultaneous $+10\%$ perturbation of masses, COM inertias, and length-dependent plant-dynamics coefficients; calibrated kinematic geometry remains fixed

Table 8: Simulation and comparison settings used for the reported final results.

Case	Parametric uncertainty	External disturbance
C1: nominal	inactive	inactive
C2: uncertainty only	10%, 1–5 s	inactive
C3: disturbance only	inactive	5–9 s
C4: combined	10%, 1–5 s	5–9 s

Table 9: Four primary evaluation scenarios.

Figure 7: Sampled joint-limit workspace and desired Cartesian reference path for the planar 3-DOF mechanism. The light background denotes the ideal geometric reach bound, the sampled points denote configurations satisfying the active joint limits, and the high-contrast curve denotes the desired path; the path is closed for display by repeating its first plotted Cartesian point without modifying the evaluated trajectory data. The plotting orientation is horizontal

Y

and vertical

X

, with 45 samples per joint.

Figure 8: Sampled cable-feasibility map for the planar 3-DOF, three-cable mechanism. Light points denote joint-limited workspace samples, the emphasized subset satisfies the nominal static gravity-torque mapping with full-rank $\bm{J}_{c}$ and nonnegative inferred cable tension, and the high-contrast curve denotes the desired Cartesian path. The path is closed for display by repeating its first plotted point without modifying saved trajectory data or performance calculations; axes are horizontal $Y$ and vertical $X$ .

Figure 9: Cable-actuation Jacobian diagnostics along the desired trajectory: minimum singular value

\sigma_{\min}(\bm{J}_{c})

, two-norm condition number

\kappa_{2}(\bm{J}_{c})

, and matrix rank. Rank remains three over the logged path, while

\sigma_{\min}

and

\kappa_{2}

identify the least favorably conditioned intervals. Light-blue and light-peach shading denote the 1–5 s uncertainty and 5–9 s disturbance intervals, respectively.

Quantity	Value
Samples per joint / total samples	45 / 91,125
Statically nonnegative-tension samples	33,921
Nonnegative-tension sample ratio	37.22%
Desired path inside geometric sample	100%
Desired path inside tested feasible sample	100%
Minimum $\sigma_{\min}(\bm{J}_{c})$ along path	0.01481

Table 10: Sampled workspace and cable-Jacobian summary for the desired trajectory.

Panel of Figure 10 — Figure 10: Cartesian tracking under combined parametric uncertainty and external disturbance. Panel (a) compares the desired path with the realized CTC and CTC + bounded residual DDPG trajectories; panel (b) reports the Euclidean Cartesian tracking-error norm. Light-blue shading marks the 1–5 s uncertainty interval and light-peach shading marks the 5–9 s disturbance interval.

Metric	CTC	CTC + residual DDPG	Change
RMS Cartesian error (m)	0.0221419	0.0128462	42.0% lower
Peak Cartesian error (m)	0.0970994	0.0782787	19.4% lower
IAE (m s)	0.156619	0.0572133	63.5% lower
ISE (m 2 s)	0.00487701	0.00162135	66.8% lower
Recovery time after disturbance onset (s)	0.08	0	–
Minimum logged cable-demand output (N)	0	3.41445	–

Table 11: Representative combined uncertainty/disturbance results from the detailed post-processing dataset.

Figure 11: Algebraically inferred cable-tension demands

F_{1}

F_{2}

, and

F_{3}

in the representative combined case. Dashed curves correspond to CTC and solid curves to CTC + bounded residual DDPG. Light-blue and light-peach shading identify the uncertainty and disturbance intervals, respectively.

Figure 12: Minimum logged algebraic cable-demand audit output,

\min_{i}F_{i}(t)

, in the representative combined case. The audit logic returns zero when its nonnegative-tension test rejects a solution, so the logged signal is nonnegative by construction. CTC reaches this zero fallback boundary, whereas the CTC + bounded residual DDPG branch retains a minimum of 3.41445 N. This diagnostic output is not applied to the simulated plant. Light-blue and light-peach shading denote the uncertainty and disturbance intervals.

Figure 13: Joint-coordinate responses

q_{1}

q_{2}

, and

q_{3}

with their active lower and upper limits in the representative combined case. The CTC trajectory contains 61 out-of-limit channel-samples, whereas the CTC + bounded residual DDPG trajectory contains none under the same sampling and tolerance convention.

Case	Controller	RMS (m)	Peak (m)	Min. tension (N)	RMS reduction
Nominal	CTC	0.016479	0.097099	0	–
Nominal	CTC + residual DDPG	0.013001	0.078279	3.7196	21.1%
Uncertainty only	CTC	0.021239	0.097099	0	–
Uncertainty only	CTC + residual DDPG	0.012875	0.078279	3.7197	39.4%
Disturbance only	CTC	0.018034	0.097099	0	–
Disturbance only	CTC + residual DDPG	0.013034	0.078279	3.9467	27.7%

Table 12: Four-case comparison from the independent case-matrix dataset.

Panel of Figure 14 — Figure 14: Cartesian tracking metrics for the nominal, uncertainty-only, disturbance-only, and combined scenarios: panel (a) reports RMS error and panel (b) reports peak error for CTC and CTC + bounded residual DDPG.

Figure 15: Integral Cartesian tracking metrics for the four scenarios. The grouped values compare IAE and ISE for CTC and CTC + bounded residual DDPG using the independently generated case-matrix dataset.

Panel of Figure 16 — Figure 16: Cartesian path and Euclidean tracking-error histories for C1 (nominal) and C2 (10% parametric uncertainty active from 1 to 5 s). Each case compares CTC with CTC + bounded residual DDPG under the same desired path and initial condition.

Panel of Figure 17 — Figure 17: Cartesian path and Euclidean tracking-error histories for C3 (external disturbance active from 5 to 9 s) and C4 (10% uncertainty from 1 to 5 s followed by disturbance from 5 to 9 s). Each case compares CTC with CTC + bounded residual DDPG.

Panel of Figure 18 — Figure 18: Ten-seed Cartesian-error summary for the four scenarios. Bars report the mean and error bars report one standard deviation for panel (a) RMS error and panel (b) peak error. Seed-dependent variability appears in the disturbance-enabled scenarios; the nominal and uncertainty-only scenarios are deterministic in this evaluation.

Figure 19: Interval-wise Cartesian RMS error averaged over ten seed runs for 0–1 s (initial transient), 1–5 s (parametric uncertainty), 5–9 s (external disturbance), and 9–10 s (post-disturbance recovery).

Case	$x_{0}$ (m)	$y_{0}$ (m)	Role
IC1	$-0.85$	$0.05$	Upper-side offset
IC2	$-0.85$	$0.00$	Mild offset
IC3	$-0.90$	$-0.05$	Nominal main-run initial state
IC4	$-0.90$	$-0.10$	Lower-side offset

Table 13: Cartesian initial conditions used in the local sensitivity test.

Figure 20: Cartesian trajectory-transfer evaluation from four nearby initial configurations under the combined uncertainty/disturbance scenario. The dotted curve is the desired trajectory, solid magenta curves are the CTC + bounded residual DDPG responses, dashed black curves are the CTC responses, and filled markers denote the four initial Cartesian positions. The plotting orientation is horizontal

Y

and vertical

X

Figure 21: Residual-action usage relative to the joint-wise limits. Panel (a) reports

\rho_{j}^{\max}=\max_{t}|\tau_{\mathrm{RL},j}|/\tau_{\mathrm{RL},j}^{\max}

. Panel (b) reports the percentage of samples satisfying

|\tau_{\mathrm{RL},j}|/\tau_{\mathrm{RL},j}^{\max}\geq 0.95

. Joint 1 reaches 0.972 and is near the bound for 0.200% of samples; joints 2 and 3 have no near-limit samples.

Figure 22: Circular-trajectory transfer evaluation under the combined uncertainty/disturbance scenario. Panel (a) compares the desired Cartesian path with the realized CTC and CTC + bounded residual DDPG trajectories; panel (b) reports the Euclidean Cartesian tracking-error norm. Light-blue and light-peach shading denote the 1–5 s uncertainty and 5–9 s disturbance intervals. RMS error decreases from 0.021738 to 0.012052 m; the associated source dataset contains 44 and 101 joint-limit channel-samples, respectively, identifying trajectory-dependent joint-coordinate coverage for subsequent constraint-aware training.

Panel of Figure 23 — Figure 23: Joint-torque components in the representative combined case. Panel (a) shows the nominal computed-torque command $\tau_{\mathrm{CTC},j}$ and panel (b) shows the bounded residual action $\tau_{\mathrm{RL},j}$ for joints $j=1,2,3$ .

Panel of Figure 24 — Figure 24: Controller command and plant disturbance in the representative combined case. Panel (a) reports the combined commanded joint torque $\bm{\tau}_{\mathrm{cmd}}=\bm{\tau}_{\mathrm{ctc}}+\bm{\tau}_{\mathrm{RL}}$ , and panel (b) reports the externally applied disturbance torque $\bm{\tau}_{d}$ . The disturbance enters the simulated plant separately from the cable-allocation command.

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