Patient-specific upper-limb robotic replica integrating joint rigidity and physiological tremor

Abstract

The implementation of an upper-limb rehabilitation simulator replicating patient-specific physical characteristics has garnered increasing interest as an effective alternative to human subjects for medical training and research. However, human upper-limb motion involves complex multi-joint coordination, whereas most existing patient simulators reproduce simplified mechanical properties of individual joints and often fail to mimic critical pathological characteristics, such as rigidity and physiological tremors. This study addresses these limitations by proposing a novel simulator that incorporates both compound joint coordination and patient-specific symptoms (e.g. lead-pipe rigidity and physiological tremors). A motion control system based on a linear quadratic regulator was designed, with patient-specific reference torques derived from adult male motion-capture data. The tremor parameters were selected within the range reported in neurophysiological literature, and rigidity was implemented based on spasticity-inspired torque models. The proposed controller achieved torque-tracking errors below 3%, while reproducing clinically relevant peak torque levels of 18–20 Nm at the shoulder and 3–7 Nm at the elbow under manual muscle testing–grade scaling. The experimental evaluations across rigidity levels highlighted the reliability and effectiveness of the proposed simulator in accurately mimicking patient characteristics, making it a practical alternative for patients in rehabilitation training. Additionally, a clinical similarity survey was conducted with eight rehabilitation specialists and 10 students following a standardized, blinded protocol, yielding a mean rating of $9.33\pm 0.72$ and a significant group effect (analysis of variance $F(2,57)=12.83$ , $p<0.001$ ). To the best of our knowledge, this study is the first to combine multi-joint coordination with the simultaneous reproduction of rigidity and physiological tremors in a two-degree-of-freedom upper-limb robotic replica, demonstrating the novelty and practical value of the approach.

Introduction

In medical training, robotic simulators replicating patient-specific physical characteristics have gained significant attention as effective substitutes for patients. Rehabilitation simulators designed to emulate spasticity or joint rigidity observed in hemiparetic patients serve as valuable training tools for rehabilitation therapists. However, most existing systems have focused on passive reproduction of joint resistance or on simplified stiffness without accounting for pathological conditions. Achieving a high degree of realism in these simulators requires advances in robotic system control to mimic patient characteristics accurately. ¹

Patient-specific replicas are critical in rehabilitation therapy, where the precise replication of limb movements can substantially enhance treatment efficacy. The ability to reproduce pathological movement patterns, such as the joint rigidity and spasticity commonly observed in hemiplegic patients, is vital for creating an effective training environment. By simulating these motor impairments, rehabilitation therapists can refine therapeutic techniques in a controlled setting that closely mimics real-world clinical conditions.^1–4

A gap exists in the realistic replication of simultaneous multi-joint interactions with pathological characteristics (e.g. tremors), which are often experienced in clinical practice. Accurate replication of joint rigidity, physiological tremors, and other abnormal motion patterns offers rehabilitation professionals invaluable opportunities for hands-on practice and clinical assessment. These simulators enable a controlled environment where therapists can hone their skills without risk to the patient. Furthermore, advanced robotic systems that simulate patient-specific impairments provide a platform for standardized and objective clinical training and evaluation (e.g. rigidity and spasticity scoring).^1,2

Robotic replicas have increasingly been focused on the demand for systems capable of simulating multi-joint and full-limb motions. Such systems are necessary to represent the coordinated motions of the shoulder, elbow, and wrist accurately, which are vital for comprehensive rehabilitation training. Furthermore, upper-limb patient simulators must incorporate varying resistance, stiffness, and abnormal muscle behavior. Several approaches have been explored to achieve a high-fidelity robotic replica, including integrating magnetorheological dampers, electronic brakes, and tension cables.^1–14

Numerous studies have explored upper-limb rehabilitation using robotic replicas that simulate joint behavior via control strategies to assist in clinical training. Takahashi et al. ⁶ developed an elbow-joint robotic replica incorporating a magnetorheological brake and direct current motor. The system featured a feedback mechanism that modulated the joint resistance torque in real time to emulate rigidity levels during physical therapy exercises. Park et al. ⁷ introduced a haptic elbow spasticity robotic replica using a cable-driven system and an electronic brake, where a proportional derivative (PD)-based feedback loop controls the cable tension to reproduce clinical spasticity profiles. Zakaria et al. ⁸ proposed a two-degree-of-freedom (two-DOF) robotic system addressing the shoulder and elbow joints. By applying impedance control, the robot replicated varying joint resistance in response to external manipulation, supporting therapist training. Liang ⁹ constructed a passive hydraulic robotic replica incorporating a Scotch-yoke structure with adjustable internal damping. In Table 1, previous research on upper-limb rigidity robotic replicas^10–14 has rarely explored control strategies, especially in multi-joint rehabilitation robots that incorporate compound multi-joint coordination and simultaneous patient-specific symptoms (e.g. lead-pipe rigidity and physiological tremors).

Table 1.

Robotic replicas replicating joint rigidity in rehabilitation patients.

Replica (DOF)	Institution	Range of movement	Expression of rigidity
Wrist (2)	Gifu University	$-90\circ$ to $90\circ$	Passive spring-damper model to emulate joint rigidity with limited active control
Elbow (1)	Shibaura Institute Technology	$0\circ$ – $135\circ$	Variable damping control via magnetorheological actuator for joint resistance modulation
Elbow (1)	KAIST	$45\circ$ – $150\circ$	Feedback-based PD control to regulate cable tension and electronic braking for spasticity
Elbow (1)	Johns Hopkins University	$0\circ$ – $143\circ$	Impedance control strategy using electronically controlled braking in a haptic interface
Elbow (1)	Saitama Prefectural University	$0\circ$ – $100\circ$	Internal passive resistance modeling using hydraulic damping for abnormal muscle response
Elbow (1)	UIUC	$45\circ$ – $165\circ$	Series elastic actuator-based compliance control for replicating lead-pipe rigidity

DOF: degree-of-freedom; PD: proportional derivative.

Integrating joint rigidity and physiological tremors is critical in developing a realistic surrogate for upper-limb rehabilitation patients. Takanokura and Sakamoto ¹⁵ analyzed the magnitude of physiological tremors in rigid shoulder, elbow, and wrist joints, and explored the relationship between tremor intensity and joint rigidity. Morrison and Newell ¹⁶ investigated the intensity of physiological tremors in static, rigid upper-limb joints. Similarly, Taheri et al. ¹⁷ explored tremor characteristics associated with the motion of a rigid wrist joint in a defined range. Despite these contributions to understanding medical symptoms, no existing research has replicated joint rigidity and physiological tremors in robotic simulators designed for rehabilitation applications. This lack of integration limits their clinical utility, especially for training that requires compound symptom expression. To bridge this gap, the present study implements an optimal control scheme for a multi-DOF upper-limb replica that accurately reproduces physiological tremor and joint rigidity, enhancing its applicability to rehabilitation training.

Mathematical model

Figure 1 presents the developed two-DOF upper-limb robotic replica. The variables $l_1$ (0.37 m) and $l_2$ (0.30 m) represent the lengths of the upper arm and forearm of an Asian male, respectively. The distances between the center of gravity of each link and the relevant joints are $l_{a1}$ (0.03 m) and $l_{a2}$ (0.02 m), respectively. ¹ The upper arm and forearm have a mass ( $m_1$ and $m_2$ ) of 2 kg and 1 kg, respectively. The rotational angle of the shoulder and elbow joint, torque, and moment of inertia are denoted as $\mathbf{q}=[q_1,q_2{]}^{\mathrm{T}}$ , $\mathit{\tau }=[{\tau }_1,{\tau }_2{]}^{\mathrm{T}}$ , and $\mathbf{I}=[I_1,I_2{]}^{\mathrm{T}}$ , respectively.

Figure 1.

Developed upper-limb robotic replica with two-degree-of-freedom (two-DOF). (a) Mathematical model and (b) real model.

The coordinates of each joint are as follows:

$$\begin{array}{cc}{\mathbf{o}}_e& =[l_1{c}_1,l_1{s}_1{]}^{\mathrm{T}},\\ {\mathbf{o}}_w& ={\mathbf{o}}_e+[l_2{c}_{12},l_2{s}_{12}{]}^{\mathrm{T}},\end{array}$$

(1)

where $s_1=\sin q_1$ , $c_1=\cos q_1$ , $s_{12}=\sin (q_1+q_2)$ , and $c_{12}=\cos (q_1+q_2)$ .

The dynamics of the developed replica are as follows:

$$\begin{array}{rl}& \left[\begin{array}{cc}1413+148c_2& 8+c_2\\ 8+c_2& 8\end{array}\right]\ddot{\mathbf{q}}-74\left[\begin{array}{c}{s}_2{\dot{q}}_2^2-2s_2{\dot{q}}_1{\dot{q}}_2\\ s_2{\dot{q}}_1^2\end{array}\right]\\ & +{10}^2\left[\begin{array}{c}43g{c}_1+2g{c}_{12}\\ 2g{c}_{12}\end{array}\right]={10}^4\mathit{\tau }.\end{array}$$

(2)

The state space form is as follows:

$$\dot{\mathbf{x}}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 12.49& -12.54& 0& 0\\ -14.49& 29.36& 0& 0\end{array}\right]\mathbf{x}+\left[\begin{array}{c}0\\ 0\\ -2.98\\ 5.98\end{array}\right]\mathbf{u},$$

(3)

$$\begin{array}{rl}y& =\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{u},\end{array}$$

where $\mathbf{C}$ represents a $4\times 4$ identity matrix, and $\mathbf{D}$ is a $4\times 1$ zero vector. Moreover, $\mathbf{x}=[{\theta }_{\mathrm{sh}},{\theta }_{\mathrm{el}},{\dot{\theta }}_{\mathrm{sh}},{\dot{\theta }}_{\mathrm{el}}{]}^{\mathrm{T}}$ , where ${\theta }_{\mathrm{sh}}$ and ${\dot{\theta }}_{\mathrm{sh}}$ are the angle and angular velocity of the shoulder, respectively. In addition, ${\theta }_{\mathrm{el}}$ and ${\dot{\theta }}_{\mathrm{el}}$ represent the angle and angular velocity of the elbow, respectively. Furthermore, $\mathbf{u}=[00{\tau }_{\text{sh-in}}{\tau }_{\text{el-in}}{]}^{\mathrm{T}}$ , where ${\tau }_{\text{sh-in}}$ and ${\tau }_{\text{el-in}}$ are the input driving torques of the shoulder and elbow, respectively. Finally, $\mathbf{y}=[00{\tau }_{\text{sh-out}}{\tau }_{\text{el-out}}{]}^{\mathrm{T}}$ , where ${\tau }_{\text{sh-out}}$ and ${\tau }_{\text{el-out}}$ denote the output driving torque of the shoulder and elbow, respectively.

Motion control

The linear quadratic regulator (LQR) scheme is applied to achieve the desired angles of the elbow and shoulder joints in the presence of multiple inputs. This scheme updates the elasticity and damping parameters in the $\mathbf{Q}$ matrix of the LQR to facilitate the desired composite joint motion in a rigid state. The diagonal elements of the $\mathbf{Q}$ matrix were assigned as weighted penalties on joint-state errors, with their coefficients scaled according to each subject-specific rigidity level, incorporating grade-dependent impedance into the tracking cost. The reference torque, ${\tau }_{\mathrm{Rig}\text{-}\mathrm{des}}$ , was subsequently determined based on the patient’s rigidity grade. The quadratic cost function, $J(t)$ , is defined as follows^17–19:

$$J(t)={\int }_0^{\mathrm{\infty }}[x^{\mathrm{\top }}(t)\mathbf{Q}x(t)+u^{\mathrm{\top }}(t)\mathbf{R}u(t)]\mathrm{d}t.$$

(4)

From equation (4), the weight matrix of the inputs, $\mathbf{R}$ , is set as an identity matrix with a value of 0.006 to achieve a fast response. The feedback gain matrix, $\mathbf{K}$ , obtained via the Riccati equation, is given in equation (5):

$$u(t)=-\mathbf{K}x(t),\mathrm{where}\mathbf{K}={\mathbf{R}}^{-1}{\mathbf{B}}^{\mathrm{\top }}\mathbf{P}.$$

(5)

The proposed LQR-based tracking controller was evaluated by reproducing upper-limb joint motion measured during three repeated trials (approximately 30 s each). The reference of the driving torque of upper-limb joints for healthy individuals, ${\tau }_{\mathrm{h}}$ , was defined based on the results by Rong et al. ²⁰ The peak torque of the shoulder and elbow joints in normal individuals is 20 and 10 Nm, respectively. The optimal feedback gain matrix, $\mathbf{K}$ , for tracking the torque of the shoulder and elbow joints in general individuals is $[129.09000;043.0600;001.190;000-3.97]$ . The effective state weighting matrix, $\mathbf{Q}$ , was adapted by scaling its diagonal elements according to grade-dependent stiffness and damping characteristics, based on Winter’s model. ²¹

Adjusting $\mathbf{Q}$ based on subject-specific joint impedance enables variability in joint behavior under pathological conditions. This controller design also supports parameter customization for individualized rehabilitation protocols. Torque-tracking errors remained within $\le$ 3%–5% across joints and grades, consistent with clinically reported peak torque ranges. Figure 2 presents the results of the simulation and experiment of the joint torque tracking of the proposed robotic replica with LQR control. In this setup, the initial and target positions for each joint on the robotic replica were set to $0\circ$ and $70\circ$ , respectively. Reference trajectories for each joint were defined using three cycles of the peak torque, and five trials were performed as presented in Figure 2. The proposed controller exhibits excellent tracking performance with maximum errors of 3% for the shoulder (Figure 2(a)) and 2% for the elbow (Figure 2(b)). At this time, the real-time control loop operated at approximately 40 Hz (25 ms cycle time). The latency was approximately 50 ms.

Figure 2.

Torque tracking of three-peak motion on the developed upper-limb robot. (a) Shoulder joint and (b) elbow joint.

Proposed control algorithm

Motion capture

This study employs motion capture to collect the trajectories of normal upper-limb motion, which are applied to the upper-limb robotic replica. Typical Asian male participants in their 30s performed a drinking motion using a cup. The starting position is with the shoulder in a neutral position, the elbow at 90 $\circ$ , the forearm in a vertical position, and the wrist in a neutral position. A cup of water is placed 30 cm away from the midline of the body. The exercise comprises three steps: moving from the initial position to the target position and returning to the initial position. The operating time for a cycle is about 8 s. To ensure consistency, the participant maintained an upright seated posture with trunk restraint, using the dominant arm only, and phase segmentation following the International Society of Biomechanics (ISB) recommendations for upper-limb kinematic analysis biomechanics. ²² Figure 3 illustrates the detailed procedural steps.

Figure 3.

Phase state of upper-limb motion.

Eleven markers were attached to anatomical landmarks according to the ISB guidelines. ²² The three-dimensional trajectories of the markers were recorded using a motion capture system (OptiTrack, Flex13) comprising six cameras calibrated at a sampling frequency of 120 Hz. The occlusion gaps of the frames ( $\le$ 10) were spline-interpolated, and the trajectories were low-pass filtered (zero-phase Butterworth). Joint coordinate systems were derived from local triads, and shoulder and elbow angles were computed using ISB-defined Euler sequences (shoulder: thorax–humerus; elbow: humerus–forearm). Figure 4 presents the variations in the shoulder and elbow angles captured during the motion experiment. During the drinking motion, the shoulder angle increased from $8\circ$ to a peak of $58\circ$ , whereas the elbow flexed from $82\circ$ to a maximum of $125\circ$ . A notable reduction in the elbow angle was observed between 2 and 3 s, indicating the braking phase of the motion.

Figure 4.

Mean of the joint angle in human upper-limb motion.

The upper-limb kinematic data collected from healthy individuals serve as input for the motion controller, which was adapted to accommodate patient-specific rigidity constraints. Although only data from one subject were used, the recorded joint angles were cross-checked against established physiological ranges reported in biomechanics literature.^15,16,23 Therefore, the motion data are regarded as representative of typical upper-limb kinematics of healthy individuals while drinking water.

Human rigidity

The proposed motion controller reproduces joint rigidity observed in hemiplegic patients by constraining the driving torque during the prescribed motion (Figure 5). The desired rigid-joint torque, ${\tau }_{\text{Rig-des}}$ , is derived from the normal joint torque, ${\tau }_{\mathrm{H}}$ , and the muscle strength grade, $H_{\mathrm{Disablelev}}$ , quantifying the patients rigidity level:

$${\tau }_{\text{Rig-des}}=H_{\mathrm{Disablelev}}\cdot {\tau }_{\mathrm{H}}.$$

(6)

Figure 5.

Block diagram of the control scheme on the robotic replica.

In equation (6), ${\tau }_{\text{Rig-des}}=[00{\tau }_{\text{rig-des-sh}}{\tau }_{\text{rig-des-el}}{]}^{\mathrm{T}}$ and ${\tau }_{\mathrm{H}}=[00{\tau }_{\text{h-sh}}{\tau }_{\text{h-el}}{]}^{\mathrm{T}}$ represent the driving torque and human torque at the shoulder and elbow joints, respectively. Moreover, ${\tau }_{\text{rig-des-sh}}$ and ${\tau }_{\text{rig-des-el}}$ represent the shoulder and elbow joint’ torques scaled according to manual muscle testing (MMT) grades, and ${\tau }_{\text{h-sh}}$ and ${\tau }_{\text{h-el}}$ correspond to the nominal torques of a healthy subject. Table 2 lists the MMT criteria.^22,24–26

Table 2.

Manual muscle testing grade.

Grade	(%)	Value	Muscle strength description
5	100	Normal	Full ROM with full resistance
4	75	Good	Full ROM against gravity with some resistance
3	50	Fair	Full ROM against gravity with no resistance
2	25	Poor	Full ROM with gravity eliminated
1	10	Trace	Slight contractility, no visible motion
0	0	Zero	No evidence of muscle contraction

ROM: range of motion.

The state-weighting matrix, $\mathbf{Q}$ , was extended to include the parameters of rigidity-related elasticity, ${\mathbf{K}}_{\mathrm{rig}}$ , and damping, ${\mathbf{C}}_{\mathrm{rig}}$ . The scaling of these parameters followed Winters’ muscle–joint model, ²¹ demonstrating that stiffness and damping increase approximately linearly with muscle tone. This relationship was mapped to clinical tone assessments by establishing a linear correspondence between modified Ashworth scale (MAS) levels and MMT grades.^22,24–26 Table 3 summarizes the resulting scaling coefficients. Each MAS-equivalent grade reduces both stiffness and damping by roughly 15% to 20%, consistent with previously reported biomechanical trends.^18,21 This clinically grounded parameterization enables the real-time adaptation of the control weighting to reflect the varying degrees of joint rigidity.

Table 3.

Scaling coefficients reflecting the correlation between MMT (strength) and MAS (tone).

MMT grade	MAS grade	Muscle tone level	${\mathbf{K}}_{\mathrm{rig}}$ (Nm/rad)	${\mathbf{C}}_{\mathrm{rig}}$ (Nms/rad)	Scaling ratio (%)
5 (normal)	0	None (baseline)	100	100	100
4	1	Slight increase in tone	85	88	85
3	2	Moderate increase	70	76	70
2	3	Marked increase	55	64	55
1	4	Severe rigidity	40	52	40

MMT: manual muscle testing; MAS: modified Ashworth scale.

To enhance physiological validity, joint torque values for healthy adults, ${\tau }_{\mathrm{H}}$ , were computed based on motion capture data. These values were scaled according to MMT grades to derive the driving torque of a rigid joint, ${\tau }_{\text{Rig-des}}$ , representing target torque levels for simulating varying degrees of joint rigidity. This method allows the robotic replica to quantify abnormal muscle tone while maintaining consistency with clinical assessment standards. The proposed framework facilitates a straightforward integration of patient-specific parameters, supporting individualized rehabilitation.

Based on the described scaled torque profiles, the effectiveness of the proposed tracking control strategy incorporating joint rigidity was evaluated on a two-DOF robotic replica via simulation and experimental validation. Figure 6(a) and (b) displays the performance of the rigidity-incorporated robotic replica under the proposed control scheme, compared with that for conventional proportional-integral-derivative (PID) and $H_{\mathrm{\infty }}$ controllers. ¹ In these figures, the black line is the reference torque for normal joint motion, and red, blue, and green lines indicate torque outputs of the robotic replica with a rigidity at Grade 3, implemented using the LQR, PID, and $H_{\mathrm{\infty }}$ controllers, respectively. In simulations of the shoulder and elbow joints, the LQR controller achieved reference-trajectory tracking with a $<$ 3% error, outperforming the PID and $H_{\mathrm{\infty }}$ schemes. The PID scheme exhibited errors exceeding 7% under the same conditions, whereas the $H_{\mathrm{\infty }}$ controller revealed intermediate performance, with root mean square (RMS) errors of approximately 5% and a slight phase lag during torque reversals. These results underscore the superior precision and disturbance tolerance of the LQR controller for upper-limb motion.

Figure 6.

Torque-tracking in LQR, PID, and $H_{\mathrm{\infty }}$ controllers in the simulation. (a) Shoulder joint (simulation, Grade 3 rigidity) and (b) elbow joint (simulation, Grade 3 rigidity). LQR: linear quadratic regulator; PID; proportional-integral-derivative.

Figure 7(a) and (b) reveals that the LQR controller tracked the reference trajectory (Grade 3) for both joints in the experiment, maintaining an error margin of 3%. In contrast, the PID controller exhibited a tracking error of approximately 5% for the same reference trajectory, while the $H_{\mathrm{\infty }}$ controller exhibited a slightly lower error of about 4.2%. Figure 8(a) and (b) represents the torque-tracking results of the two-DOF upper-limb robotic replica in the experiment. Although tracking errors increased due to the physical characteristics of the robot platform, the experimental results were closely aligned with the simulation outcomes, confirming the robustness of the proposed control approach. These findings indicate that the LQR controller is suitable for tracking complex joint torque trajectories in upper-limb motion.

Figure 7.

Torque-tracking in LQR, PID, and $H_{\mathrm{\infty }}$ controllers in the experiment. (a) Shoulder joint (experiment, Grade 3 rigidity) and (b) elbow joint (experiment, Grade 3 rigidity). LQR: linear quadratic regulator; PID; proportional-integral-derivative.

Figure 8.

Torque-tracking results by rigidity level corresponding to MMT–MAS mapping (MMT Grades 1–3). (a) Shoulder joint torque tracking by rigidity level and (b) elbow joint torque tracking by rigidity level. MMT: manual muscle testing; MAS: modified Ashworth scale.

Figure 8 displays the torque tracking of the LQR controller for upper-limb joints with varying rigidity levels, corresponding to MMT Grades 1 to 3. For the shoulder joint, rigidity at MMT Grade 3 resulted in a peak torque of 13 Nm at 40 s, whereas MMT Grades 2 and 1 produced peak torques of 10 Nm at 41 s and 6.5 Nm at 39 s, respectively. For the elbow joint, rigidity at MMT Grades 3, 2, and 1 resulted in peak torques of 5.8 Nm at 47 s, 4.1 Nm at 45 s, and 2.7 Nm at 43 s, respectively. These results demonstrate the LQR controller’s capability of tracking torque trajectories effectively across rigidity levels in the shoulder and elbow joints.

Physiological tremor

Patients with hemiparesis may experience tremors when subjected to resistance exceeding the desired joint torque during rehabilitation exercises.^15–17 To replicate the tremor characteristics, this study incorporates a physiological tremor, ${\mathbf{H}}_{\mathrm{Tr}}$ , into the proposed robotic replica. This tremor is designed based on the research findings by Takanokura and Sakamoto ¹⁵ and Zakaria et al. ⁸ The damping coefficient, ${\mathbf{C}}_{{\mathbf{H}}_{\mathrm{Tr}}}$ , and spring constant, ${\mathbf{K}}_{{\mathbf{H}}_{\mathrm{Tr}}}$ , were selected to consider tremor dynamics. The muscle mass constant, $H_{\mathrm{Mu}}$ , was determined within a range of $\pm$ 0.5, based on the average values reported for East Asian males.

The proposed model enables the robotic replica to reproduce involuntary oscillatory responses, providing a platform for evaluating tremor-suppression strategies in robotic rehabilitation. The time-dependent oscillatory torque patterns are characteristic of upper motor neuron lesions and integrate rigidity and tremor-related parameters into a unified representation of spastic resistance and involuntary joint instability. Mechanical parameters were scaled according to the MMT grades to reflect subject-specific impairments. For example, in MMT Grade 3, the forearm spring constant ( ${\mathbf{K}}_{\mathrm{rigf}}$ ) was reduced by 25% compared to the normal muscle spring constant( ${\mathbf{K}}_{\mathrm{hf}}$ ), whereas the damping coefficient( ${\mathbf{C}}_{\mathrm{rigf}}$ ) was increased by 40%, consistent with clinically observed features of moderate rigidity. The physiological tremor is modeled in equation (7) as the scaled difference between the joint responses in healthy and spastic conditions. Each term reflects physiologically grounded parameters. The spring constant, ${\mathbf{K}}_{\mathrm{H}}$ , and damping coefficient, ${\mathbf{C}}_{\mathrm{H}}$ , were derived from reported clinical data: healthy muscle rigidity typically ranges from 0.7 to 1.2 Nm/deg, and damping from 0.3 to 0.8 Nm $\cdot$ s/deg. ²¹ In patients with MMT Grade 3, these values are typically altered by a 40% increase in damping and a 25% decrease in rigidity, in line with known neuromuscular deficits.^15–17 The resulting model output yielded oscillatory profiles with tremor frequencies in the range of 4–7 Hz and amplitudes between 0.4 and 0.6 Nm, closely matching electromyography (EMG)-based tremor measurements reported in previous literature.^15–17

The physiological-tremor transfer function ${\mathbf{H}}_{\mathrm{Tr}}$ is defined as follows^{15–17,21,24}:

$$\begin{array}{rl}{\mathbf{H}}_{\mathrm{Tr}}& =H_{\mathrm{Mu}}({\mathbf{e}}_{\tau }{\mathbf{H}}_{\mathbf{I}}-{\mathit{\tau }}_{\mathbf{T}\mathbf{r}\mathbf{e}\mathbf{m}}{\mathbf{H}}_{\mathbf{I}})\\ & =H_{\mathrm{Mu}}(({\mathbf{M}}_{\mathrm{hu}}-{\mathbf{M}}_{\mathrm{trem}}){\ddot{\mathbf{e}}}_{\theta }+({\mathbf{C}}_{\mathrm{hu}}-{\mathbf{C}}_{\mathrm{trem}}){\dot{\mathbf{e}}}_{\theta }\\ & +({\mathbf{K}}_{\mathrm{hu}}-{\mathbf{K}}_{\mathrm{trem}}){\mathbf{e}}_{\theta }){\mathbf{H}}_{\mathbf{I}}.\end{array}$$

(7)

In equation (7), ${\mathbf{e}}_{\tau }$ and ${\mathit{\tau }}_{\mathbf{T}\mathbf{r}}$ denote the torque error of each joint and the tremor torque specific to hemiplegic patients, respectively. Moreover, ${\mathbf{M}}_{\mathrm{hu}}$ and ${\mathbf{M}}_{\mathrm{trem}}$ represent the muscle mass matrices for a healthy and a hemiplegic patient, denoted as $\mathrm{diag}(0,0,m_{\mathrm{hu}},m_{\mathrm{hf}})$ and $\mathrm{diag}(0,0,m_{\mathrm{tremu}},m_{\mathrm{tremf}})$ . Further, $m_{\mathrm{hu}}$ and $m_{\mathrm{tremu}}$ correspond to the upper arm muscle mass of a healthy and a hemiplegic patient, respectively, whereas $m_{\mathrm{hf}}$ and $m_{\mathrm{tremf}}$ are the muscle mass of the forearm for a healthy and a hemiplegic patient. Additionally, ${\mathbf{C}}_{\mathrm{hu}}$ and ${\mathbf{C}}_{\mathrm{Trem}}$ denote the damping matrices for a healthy and a hemiplegic muscles, as $\mathrm{diag}(0,0,c_{\mathrm{hu}},c_{\mathrm{hf}})$ and $\mathrm{diag}(0,0,c_{\mathrm{tremu}},c_{\mathrm{tremf}})$ , respectively. The variables $c_{\mathrm{hu}}$ and $c_{\mathrm{tremu}}$ denote the damping coefficients of the upper arm muscles for a healthy and hemiplegic patient, whereas $c_{\mathrm{hf}}$ and $c_{\mathrm{tremf}}$ represent the respective damping coefficients of the forearm muscles. In addition, ${\mathbf{K}}_{\mathrm{hu}}(=\mathrm{diag}(k_{\mathrm{hu}},k_f,0,0))$ and ${\mathbf{K}}_{\mathrm{Trem}}(=\mathrm{diag}(k_{\mathrm{tremu}},k_{\mathrm{tremf}},0,0)$ ) represent the spring matrices for the muscles of a healthy and a hemiplegic patient, respectively. The variables $k_{\mathrm{hu}}$ and $k_{\mathrm{tremu}}$ indicate the spring coefficients of the upper arm muscles for an individual and hemiplegic patient, whereas $k_{\mathrm{hf}}$ and $k_{\mathrm{tremf}}$ represent the respective spring coefficients of the forearm muscles. Finally, ${\mathbf{H}}_{\mathbf{I}}$ serves as the unit matrix introduced to align the matrix structure of ${\mathbf{H}}_{\mathrm{Tr}}$ with that of the $\mathbf{Q}$ matrix.

Figure 9 presents the computed physiological-tremor result for the shoulder and elbow joints under rigidity conditions corresponding to MMT Grade 3 (dominant tremor frequency: the 4–7 Hz). During 100 s of complex movement, the shoulder joint (Figure 9(a)) exhibits an average physiological-tremor amplitude of 0.21 Nm, with a peak tremor of 0.58 Nm. Similarly, the elbow joint (Figure 9(b)) demonstrates an average tremor amplitude of 0.17 Nm, with a maximum tremor of 0.44 Nm. These results with the parameterized stiffness and damping adjustments ( $-$ 25% stiffness, +40% damping at Grade 3) align with previously reported physiological tremor ranges.

Figure 9.

Multiple rigid joints with physiological tremor (manual muscle testing (MMT) Grade 3) in the simulation. (a) Shoulder joint and (b) elbow joint.

Integrated control algorithm

The tremor component is incorporated into the two-DOF upper-limb robotic replica by reformulating the previously computed matrix, ${\mathbf{H}}_{\mathrm{Tr}}$ , as $\mathbf{Q}$ , enabling a joint-level tremor representation within the proposed system. The resulting transformation is updated to $Q_{\mathrm{up}}$ , as defined in equation (8):

$${\mathbf{Q}}_{\mathrm{up}}=H_{\mathrm{Mu}}\left[\begin{array}{cc}({\mathbf{K}}_{\mathrm{hu}}-{\mathbf{K}}_{\mathrm{Trem}}){\mathbf{e}}_{\theta }& ({\mathbf{C}}_{\mathrm{hu}}-{\mathbf{C}}_{\mathrm{Trem}})\dot{{\mathbf{e}}_{\theta }}\\ {\mathbf{0}}_{2\times 2}& ({\mathbf{M}}_{\mathrm{hu}}-{\mathbf{M}}_{\mathrm{trem}})\ddot{{\mathbf{e}}_{\theta }}\end{array}\right]{\mathbf{H}}_{\mathbf{I}}.$$

(8)

By adjusting the elastic and damping forces based on an individual’s muscle mass, the rigid joint of the two DOF upper-limb robotic replica can replicate changes in muscle elasticity and damping characteristics observed in rehabilitation patients. Figure 10 presents the implementation of complex rigid-joint motions incorporating physiological tremors in the proposed robotic replica. The experimental trials were conducted with a driving torque corresponding to MMT Grade 3, where tremor inputs were applied from the initial posture to the target position of each joint. The experiments were designed to evaluate peak torque tracking performance under compound upper-limb motion.

Figure 10.

Proposed control system architecture of the two-degree-of-freedom (two-DOF) upper-limb robotic replica.

Figure 11 presents the torque-tracking performance of the three controllers at the shoulder and elbow joints. Quantitatively, the RMS tracking errors for LQR control in Figure 11(a) and (b), PID control in Figure 11(c) and (d), and $H_{\mathrm{\infty }}$ control in Figure 11(e) and (f) were 2.7%, 6.1%, and 4.0% at the shoulder joint, and 3.1%, 6.4%, and 4.5% at the elbow joint, respectively. Across all controllers, the peak-to-peak tremor amplitudes remained within $\pm 0.2$ of the reference, confirming the preservation of the intended 4–7 Hz tremor band. These findings indicate that the proposed LQR scheme offers the most clinically realistic tremor reproduction with minimal distortion, whereas PID exhibits pronounced sensitivity to sensor noise, and $H_{\mathrm{\infty }}$ achieves disturbance robustness with amplitude attenuation.

Figure 11.

Torque tracking with physiological tremor (MMT Grade 3) for LQR, PID, and $H_{\mathrm{\infty }}$ controllers. (a) LQR output at the shoulder, (b) LQR output at the elbow, (c) PID output at the shoulder, (d) PID output at the elbow, (e) $H_{\mathrm{\infty }}$ output at the shoulder, and (f) $H_{\mathrm{\infty }}$ output at the elbow. MMT: manual muscle testing; LQR: linear quadratic regulator; PID; proportional-integral-derivative.

To evaluate the performance of the proposed controller further across varying joint rigidity conditions, experimental trials were conducted by MMT grade. The shoulder joint was assessed under MMT Grades 4 and 2. Figure 12(a) and (b) illustrates the effective performance of the controller in simulating joint behavior across rigidity levels, validating its adaptability to various patient conditions. Consistent with previous experimental results, the shoulder joint, rigidified to MMT Grade 4, and the elbow joint, rigidified to MMT Grade 2, followed the reference trajectory with a tracking error of <5%.

Figure 12.

Torque tracking of multiple rigid joints by rigidity condition. (a) Shoulder joint (manual muscle testing (MMT) Grade 4) and (b) elbow joint (MMT Grade 2).

To substantiate the physiological-tremor behavior observed in the experiments, rigidity modeling was quantitatively parameterized according to clinically defined MMT grades. The inertial ( $M_{\mathrm{Rig}}$ ), damping coefficients ( ${\mathbf{C}}_{\mathrm{Tr}}$ ), and spring constant ( ${\mathbf{K}}_{\mathrm{Tr}}$ ) matrices were systematically scaled to reflect varying muscle tone levels. As the MMT grade decreased from Levels 4 to 2, the spring constants ( $k_{\mathrm{tr}.\mathrm{u}},k_{\mathrm{tr}.\mathrm{f}}$ ) and damping coefficients ( $c_{\mathrm{tr}.\mathrm{u}},c_{\mathrm{tr}.\mathrm{f}}$ ) were linearly reduced by approximately 25% per grade level, representing progressive loss in voluntary motor control and joint stability. These parameter adjustments were selected to replicate the EMG-derived tremor characteristics reported in the literature, particularly within the frequency range of 4–12 Hz and angular displacements between $5^{\circ }$ and ${25}^{\circ }$ .^8,15

The torque profiles in Figures 10 and 11 reflect the joint responses under the rigidity-adjusted conditions and confirm the validity of the selected tremor parameters. For instance, in the shoulder joint modeled with MMT Grade 3 characteristics ( $k_{\mathrm{tr}.\mathrm{u}}=18.5$  N/m, $c_{\mathrm{tr}.\mathrm{u}}=0.75$  Ns/m), the peak oscillatory torque reached approximately 20 Nm at 18 s, closely aligning with clinically observed tremor magnitudes in post-stroke patients.^24–26 These findings further validate its capacity to emulate patient-specific tremor responses across a range of motor impairment levels.

The integrated LQR-based control framework establishes a robust foundation for replicating upper-limb tremor and rigidity phenomena in rehabilitation robots. The proposed control method presented the most stable tradeoff between tracking fidelity and physiological realism, offering clinically relevant torque outputs with RMS errors below 3%.

Survey

A satisfaction survey was conducted to evaluate the similarity of the developed upper-limb robotic replica that incorporates joint rigidity to patient movement. The evaluators comprised a group of rehabilitation professionals, including eight therapists and 10 students from the therapeutic department. The participants assessed the robotic replica using a 10-point similarity scale, ranging from 1 (not similar at all) to 10 (very similar). Prior to the evaluation, the functionality of the upper-limb robotic replica was explained to the participants. After testing the degree of joint rigidity in the robotic replica, the participants completed the questionnaire to provide assessments (KNU Institutional Review Board No. 2024-0517).

A standardized protocol was implemented to enhance the methodological rigor of the similarity evaluation. Prior to assessment, participants underwent a brief exposure protocol consisting of 2 min of passive mobilization with an actual patient exhibiting upper-limb rigidity, followed by 2 min of interaction with the developed robotic replica; exposure order was randomized (patient-first vs. robot-first), and evaluators were blinded to the controller type. The order of exposure was randomized, and evaluator blinding was applied to minimize potential bias. The concept of similarity was assessed across three distinct domains: joint rigidity, tremor rhythmicity, and overall movement impression. Each domain was rated on a 5-point Likert scale, and the average score was normalized to a 10-point scale for consistency with the previous literature. Fleiss’s kappa coefficient was calculated to verify inter-rater reliability based on evaluations from four physicians selected from the participant group. These therapists, each with substantial clinical experience in neurorehabilitation, independently assessed all experimental conditions following the standardized evaluation protocol. The resulting Fleiss kappa value was 0.73, corresponding to substantial agreement according to conventional criteria.^27,28

The objective of this evaluation was to assess the similarity between composite rigid joint movements in upper-limb rehabilitation robots and those observed in patients. The evaluation is categorized into three groups: Group_PID refers to the application of a PID controller, Group LQR refers to the application of an LQR controller, and Group_TR refers to the application of an LQR controller with physiological tremor. This classification enables a comparative analysis of the control strategies in replicating patient-specific joint movements with rigidity.

Figure 13 reveals that the rigid joint movements of the upper-limb robotic replica controlled by the PID controller were rated with an average similarity score of 7.25 ( $\pm$ 0.94). For the LQR controller, the similarity rating was higher, with an average score of 8.33 ( $\pm$ 0.88). The highest similarity was achieved when physiological tremors were included in the LQR controller, scoring an average of 9.33 ( $\pm$ 0.72). These results indicate that incorporating physiological tremors into the control strategy leads to an increase in perceived similarity to patient-specific upper-limb movements.

Figure 13.

Similarity survey on joint rigidity between a robotic replica and patient.

A one-way analysis of variance was conducted to determine the effect of the controller type on the perceived similarity.^29,30 This classification enables a comparative analysis of control strategies in replicating patient-specific joint movements with rigidity. The results indicated a statistically significant main effect, $F(2,57)=12.83$ , $p<0.001$ . Post-hoc comparisons using Tukey’s honest significant difference test ³⁰ indicated that Group_TR (LQR with tremor) received significantly higher similarity ratings than Group_PID ( $p<0.001$ ) and Group_LQR ( $p=0.031$ ), and that Group_LQR also outperformed Group_PID ( $p=0.047$ ). Effect sizes were Cohen’s $d=2.49$ (Group_TR vs. Group_PID) and $d=1.24$ (Group_TR vs. Group_LQR), both falling in the large-effect range according to conventional benchmarks. ³¹ These findings indicate that including physiological tremors in the control strategy statistically significantly enhances the perceived fidelity of simulated upper-limb movement.

Conclusion

This study presents an LQR-based control approach for implementing complex rigid joint movements in a two-DOF upper-limb patient robotic replica, incorporating physiological tremors in the shoulder and elbow joints. The joint driving torque was computed based on the motion capture data for the upper-limb motion of an adult male. Torque tracking across varying rigidity levels by joint maintains an error of $<$ 3%, confirming that the complex rigid joint motion in the proposed robotic replica, including physiological tremors, closely resembles human motion.

The evaluations were conducted to assess the clinical relevance and accuracy of the proposed robotic replica. A similarity assessment involving rehabilitation experts revealed that the LQR controller incorporating physiological tremor dynamics achieved the highest similarity scores for patient-specific joint behavior. The mean similarity ratings were $7.25\pm 0.94$ for PID, $8.33\pm 0.88$ for LQR, and $9.33\pm 0.72$ for LQR with tremors, displaying a clear monotonic increase in perceived clinical realism across the three controllers. Inter-rater consistency was quantified using Fleiss’ kappa, yielding a value of 0.73, indicating substantial agreement between evaluators.

To replicate patient-specific torque characteristics, the upper-limb rehabilitation robotic replica integrates joint stiffness, damping properties, and physiological-tremor parameters to emulate varying levels of motor impairment. These parameters were linearly scaled by combining Winters’ muscle-tone–strength proportionality model with MMT grading, establishing an initial modeling framework that concurrently reflects increased muscle tone and reduced voluntary strength. Nevertheless, this linear scaling approach cannot fully capture time-dependent viscoelastic changes or nonlinear fatigue effects in musculoskeletal dynamics. Therefore, future work should extend this framework to an adaptive control architecture in which stiffness and damping parameters are continuously updated in real time using EMG-based feedback from muscle activation.