2. System modeling

We utilize the Lagrangian method, which is based on energy analysis, to find the dynamic equation of the model. Each leg of the lower-limb exoskeleton is represented by a 2-DOF mechanism, in which its hip and knee joints are flexibly actuated in the sagittal plane as shown in Figure 1. The uncertainty of each flexible actuated joint is modeled by the displacement between the flexible joint and any point on the rigid frame as

(1) $$ {y}_i\left({x}_i,t\right)=\sum \limits_{j=1}^n\;{\xi}_{ij}(t){\phi}_{ij}(t), $$

where $ {\xi}_{ij}(t) $ is time-dependent coordinates of the joint and $ {\phi}_{ij}(t) $ is the spatially dependent $ {i}^{\mathrm{th}} $ mode shape function of link $ j $ (Rahimi and Nazemizadeh, Reference Rahimi and Nazemizadeh2014; Shalaby, Reference Shalaby2018). In our exoskeleton model, the links are fixed-mass beams that are flexible in two directions within the sagittal plane, that is, each has two modes of vibration. Their mode shapes were defined using their boundary conditions. The mode shapes were normalized to a magnitude of 1 (i.e., unit vector) to simplify the final model.

Figure 1.

Biomechanical model of an exoskeleton-assisted walking system in the sagittal plane, showing the exoskeleton’s structure and its interaction with human joints. The dynamic model uses the hip ( $ {q}_1 $ ) and knee ( $ {q}_2 $ ) joint angles as generalized coordinates to illustrate the relationship among exoskeleton angular position, joint stiffness, damping, and human joint behavior.

For the case of two modes of vibration for each link, the kinetic and potential energy expressions for the two joints, the two links, and the links’ mass were derived using the kinematics relationships between the segments. The final equations of motion were then derived using Lagrange energy techniques and coincided with those presented by De Luca et al. (Reference De Luca, Lanari and Ulivi1991). The model of the system is represented as follows:

(2) $$ M(q)\ddot{q}+B\left(q,\dot{q}\right)+K(q)=Q, $$

where $ q={\left[{q}_1,{q}_2\right]}^T $ is the generalized coordinate vector, $ {q}_1 $ and $ {q}_2 $ are the hip and knee angle (Figure 1), $ M\in {\mathcal{R}}^m $ is the inertia matrix, $ B\left(q,\dot{q}\right) $ is the vector of Coriolis centrifugal forces and the motor and structural damping terms, $ k $ is the stiffness matrix, and $ Q $ is the vector of generalized forces.

In this work, we assume that the flexible links move in a plane where the gravitational effect exists, and the flexible links are rigidly coupled to two actuated joints. The exoskeleton utilizes brushless DC (BLDC) motors to actuate the joints that are modeled as

(3) $$ {\displaystyle \begin{array}{c}{v}_a={R}_p{i}_a+L\frac{d{i}_a}{dt}+{E}_a,\\ {}{v}_b={R}_p{i}_b+L\frac{d{i}_b}{dt}+{E}_b,\\ {}{v}_c={R}_p{i}_c+L\frac{d{i}_c}{dt}+{E}_c,\end{array}} $$

where a, b, and c are motor windings, $ v $ is the phase voltage (V), $ i $ is the phase current (A), $ E $ is the back electromotive force (EMF), $ {R}_p $ is the phase resistance in ( $ \Omega $ ), and $ L $ is the phase inductance (H). The ultimate BLDC motor torque, derived from the previous equation, is as in Lund et al. (Reference Lund, Billeschou and Larsen2019):

(4) $$ {T}_e={k}_f{\omega}_m+J\frac{d{\omega}_m}{dt}+{T}_L, $$

where $ {T}_e $ is the generated electrical torque (Nm), $ {T}_L $ is the mechanical load, $ J $ is the rotor inertia (kg m²), $ {k}_f $ is the frictional constant (Nm/rad), and $ \omega $ is the rotor speed (rad/s).

For each joint, equation (4) estimates the joint torques in the Lagrange formulation of the exoskeleton equations of motion, considering $ Q $ in equation (2) as

(5) $$ {Q}_i={T}_{L_i}-{nT}_{e_i}, $$

where $ i\in \mathrm{R} $ is the number of exoskeleton revolute joints.

3. Adaptive control design

To derive our adaptive control law, we utilized a previously developed analytical model of a two-link, two-actuator system, assuming clamped boundary conditions and two vibration modes per link as in Mutti et al. (Reference Mutti, Du and Nair1998) and Zhang et al. (Reference Zhang, Xu, Nair and Chellaboina2005).

The goal of the adaptive control law is to precisely control the joint angles to track the desired trajectories in the presence of system uncertainty. Taking into account the system uncertainties produced by imprecise modeling, such as system parameter perturbation, external disturbances, and non-ideal boundary conditions, the dynamic equations of the two flexible link system, as formulated in equation (3), can be rewritten as

(6) $$ M(q)\ddot{q}+B\left(q,\dot{q}\right)+K(q)=Q+{\Delta}^{\prime }, $$

where $ M(q) $ is the inertia matrix, $ \ddot{q} $ is the acceleration vector, $ B\left(q,\dot{q}\right) $ is the Coriolis and centripetal forces, $ K(q) $ is the gravitational forces, $ Q $ is the generalized forces, and $ {\Delta}^{\prime } $ is the uncertainties vector with its components being continuous functions of $ q $ and $ \dot{q} $ , encompassing modeling errors, external perturbations, and unmodeled dynamics. The essential properties of equation (6) are defined as

Property (1): The inertial matrix $ M(q) $ is a symmetric positive-definite matrix and satisfies $ {m}_oI\le M(q)\le \overline{m}I $ , where $ {m}_0,\overline{m}\in {\mathrm{\mathbb{R}}}^{+} $ and $ I $ is the identify matrix;

Property (2): The matrix $ \dot{M}(q)-2C\left(q,\dot{q}\right) $ is skew-symmetric such that $ {\gamma}^T\left(\dot{M}(q)-2C\left(q,\dot{q}\right)\right)\gamma =0 $ , where $ \gamma $ is any real vector such that $ \forall \gamma, q,\dot{q}\in {\mathrm{\mathbb{R}}}^n $ ;

Property (3): The dynamics of the links are linearly parameterized such that $ \forall x,y\in \mathrm{\mathbb{R}} $ ,

$$ M(q)+2C\left(q,\dot{q}\right)+G(q)=Y\left(q,\dot{q},x,y\right)\theta +\tau (t), $$

where $ Y\left(q,\dot{q},x,y\right)\theta \in {\mathrm{\mathbb{R}}}^{n\times p} $ is a regressor matrix and $ \theta \in {\mathrm{\mathbb{R}}}^{p\times 1} $ are constants with high uncertainty, such as exoskeleton physical parameters like link mass, moments of inertia, or other dynamic properties. Collectively, this term represents joint displacement due to uncertainty, as in equation (1). Additionally, $ M(q)\in {\mathrm{\mathbb{R}}}^{n\times n} $ is the positive-definite inertia matrix, $ C\left(q,\dot{q}\right)\in {\mathrm{\mathbb{R}}}^{n\times n} $ is the Coriolis and centripetal torque matrix, $ G(q)\in {\mathrm{\mathbb{R}}}^n $ denotes the gravitational force, and $ \tau (t)\in {\mathrm{\mathbb{R}}}^n $ denotes the control torque. We can then reformulate equation (6) by decomposing into separate rigid and flexible modes:

(7) $$ \left[\begin{array}{ll}{M}_{11}& {M}_{12}\\ {}{M}_{21}& {M}_{22}\end{array}\right]\left[\begin{array}{l}{\ddot{q}}_r\\ {}{\ddot{q}}_f\end{array}\right]+\left[\begin{array}{l}{F}_1\\ {}{F}_2\end{array}\right]=\left[\begin{array}{l}{Q}_r\\ {}0\end{array}\right]+\left[\begin{array}{l}{\Delta}_1^{\prime}\\ {}{\Delta}_2^{\prime}\end{array}\right], $$

where $ {Q}_r $ is the vector of intrinsic and external rigid torques, $ F=B\left(q,\dot{q}\right)+K(q) $ , rigid angle vector $ {q}_r={\left[{\theta}_1,{\theta}_2\right]}^T $ , and flexible angle vector $ {q}_f={\left[{\delta}_1,{\delta}_2\right]}^T $ . From equation (7), we can define

(8) $$ {\ddot{q}}_f=-{M}_{22}^{-1}\left({M}_{21}{\ddot{q}}_r+{F}_2\right)+{\Delta}_2^{{\prime\prime} }, $$

where $ {\Delta}_2^{{\prime\prime} }={M}_{22}^{-1}{\Delta}_2^{\prime } $ . Substituting (8) into the lower row of (7), it follows that

(9) $$ {\displaystyle \begin{array}{r}{M}_{11}{\ddot{q}}_r-{M}_{12}\left({M}_{22}^{-1}{M}_{21}{\ddot{q}}_r+{M}_{22}^{-1}{F}_2+{\Delta}_2^{{\prime\prime}}\right)+{F}_1={Q}_r+{\Delta}_1^{\prime}\end{array}}, $$

(10) $$ {\displaystyle \begin{array}{l}\left({M}_{11}-{M}_{12}{M}_{22}^{-1}{M}_{21}\right){\ddot{q}}_r+\left({F}_1-{M}_{12}{M}_{22}^{-1}{F}_2\right)={Q}_r+{\Delta}_1+{\Delta}_2^{{\prime\prime}}\end{array}}, $$

(11) $$ {\displaystyle \begin{array}{r}\left({M}_{11}-{M}_{12}{M}_{22}^{-1}{M}_{21}\right){\ddot{q}}_r+\left({F}_1-{M}_{12}{M}_{22}^{-1}{F}_2\right)={Q}_r+{\alpha}^{\prime}\end{array}}, $$

where $ {\Delta}_2^{\prime \prime \prime }={M}_{12}{\Delta}_2^{{\prime\prime} } $ and $ {\alpha}^{\prime }={\Delta}_1^{\prime }+{\Delta}_2^{\prime \prime \prime } $ are the vectors quantifying system uncertainty as unknown continuous functions of $ q $ and $ \dot{q} $ . Because $ {M}_{11}-{M}_{12}{M}_{22}^{-1}{M}_{21} $ is nonsingular everywhere in motion space, $ {\alpha}^{\prime } $ can be written as

(12) $$ {\alpha}^{\prime }=\left({M}_{11}-{M}_{12}{M}_{22}^{-1}{M}_{21}\right)\alpha $$

Let $ x={\left[q,\dot{q}\right]}^T $ and $ \alpha ={\left[{\alpha}_1,{\alpha}_2\right]}^T $ . Because $ {\alpha}_1 $ and $ {\alpha}_2 $ are continuous functions in $ q $ and $ \dot{q} $ , $ \alpha $ can be approximated using a Gaussian regressor (more details can be found in Sanner and Slotine, Reference Sanner and Slotine1992) with $ x $ being the input vector, such as

(13) $$ \alpha =\left[\begin{array}{c}{\mathbf{c}}_1^{\ast \mathrm{T}}\mathbf{g}(x)\\ {}{\mathbf{c}}_2^{\ast \mathrm{T}}\mathbf{g}(x)\end{array}\right], $$

where $ {\boldsymbol{c}}_i^{\ast }={\left[{c}_{i,1}^{\ast }{c}_{i,2}^{\ast}\cdots {c}_{i,m}^{\ast}\right]}^T,\hskip0.3em i=1,2\dots m $ , are the output weight vectors, $ m $ is the number of nodes in the Gaussian regressor, and the Gaussian basis function vector is selected as

(14) $$ \mathbf{g}(x)=\left[{e}^{-\frac{\mid x-{x}_1{\parallel}_2^2}{\sigma_1^2}}{e}^{-\frac{{\left\Vert x-{x}_2\right\Vert}_2^2}{\sigma_2^2}}\cdots {e}^{-\frac{{\left\Vert x-{x}_m\right\Vert}_2^2}{\sigma_m^2}}\right], $$

where $ x={\left[q,\dot{q}\right]}^T $ and $ {x}_i $ and $ {\sigma}_j^2 $ , $ j=1\dots m $ , are the center points and the variances, respectively. Using equation (7) to compute joint torques $ u $ , we let

(15) $$ {Q}_r=\left({M}_{11}-{M}_{12}{M}_{22}^{-1}{M}_{21}\right)u+\left({F}_1-{M}_{12}{M}_{22}^{-1}{F}_2\right). $$

The joint torques $ u $ can then be calculated as

(16) $$ u={\ddot{q}}_{rd}-{K}_d\left({\dot{q}}_r-{\dot{q}}_{rd}\right)-{K}_p\left({q}_r-{q}_{rd}\right)-\left[\begin{array}{l}{\mathbf{c}}_1^T\mathbf{g}(x)\\ {}{\mathbf{c}}_2^T\mathbf{g}(x)\end{array}\right], $$

where $ {\boldsymbol{c}}_i={\left[{c}_{i,1}{c}_{i,2}\cdots {c}_{i,m}\right]}^{\mathrm{T}}, $ and the joint index and Gaussian node index are $ i $ and $ m $ , respectively. If the desired joint angle is $ {q}_{rd} $ , the system dynamics with $ {e}_r={q}_r-{q}_{rd} $ and $ \Delta {c}_i={c}_i^{\ast }-{c}_i $ become

(17) $$ {\ddot{e}}_r+{K}_d{\dot{e}}_r+{K}_p{e}_r=\left[\begin{array}{c}\Delta {\mathbf{c}}_1^T\mathbf{g}(x)\\ {}\Delta {\mathbf{c}}_2^T\mathbf{g}(x)\end{array}\right], $$

(18) $$ {\ddot{e}}_{r,i}+{k}_{d,i}{\dot{e}}_{r,i}+{k}_{p,i}{e}_{r,i}=\Delta {\mathbf{c}}_i^T\mathbf{g}(x), $$

where $ {K}_d $ and $ {K}_p $ are selected as diagonal matrices. To achieve stable performance, the feedback gains of equation (17) are selected with negative real roots, such as

(19) $$ {s}^2+{k}_{d,j}s+{k}_{p,i}=\left(s+{\lambda}_{r,i1}\right)\left(s+{\lambda}_{r,i2}\right),\hskip1em i=1,2. $$

If we consider the update mechanism output as

(20) $$ {\delta}_i(t)={\dot{e}}_{r,i}+{\lambda}_{r,i}{e}_{r,i}, $$

then we can show that

(21) $$ {\dot{\delta}}_i(t)=-{\lambda}_{r,i}{\delta}_i(t)+\Delta {\mathbf{c}}_i^T\mathbf{g}(x), $$

where $ {\lambda}_{r,i} $ is selected to achieve a reasonable convergence $ {\delta}_i(t)\to 0 $ or $ {e}_{r,i}(t)\to 0 $ as $ t\to \infty $ .

The adaptive law for this system can then be formulated as

(22) $$ \Delta {\dot{\mathbf{c}}}_i=-{\eta \delta}_i(t)\mathbf{g}(x),\hskip1em i=1,2, $$

where $ \eta \in \left[0,1\right]\subset \mathrm{\mathbb{R}} $ is the adaptation gain that governs the rate of the adaptation process. Additional information regarding the asymptotic stability of this update law can be found in the Appendix. In order to compute the torque vector needed to track a desired trajectory based on the Lagrangian dynamic equation of motion, a fixed number of Gaussian nodes is selected to form the adaptation law. In this work, we utilized five nodes. For additional details on the Gaussian node structure, please refer to Sanner and Slotine (Reference Sanner and Slotine1992). The update law is calculated by local integration to correct the computed torque values iteratively as in equation (16). We call this control scheme a Gaussian-based adaptive control (GBAC).

4. Control assessment

Here, we utilized a simulation evaluation to compare the performance of our novel proposed GBAC control with a robust adaptive backstepping (BS) controller that adaptively updates the unknown dynamic exoskeleton parameters derived from Lagrange mechanics, and was previously deployed in a trajectory tracking task for a lower extremity exoskeleton (Narayan et al., Reference Narayan, Abbas and Dwivedy2023). We also implemented a well-known adaptive control scheme derived based on Lyapunov stability (Slotine–Li; Slotine and Weiping, Reference Slotine and Weiping1988) and a PID controller as baseline comparators. The simulation study evaluates algorithm tracking and convergence for a range of exoskeleton user limb masses and lengths, which represent practical sources of parametric uncertainty in pediatric populations who experience periods of rapid growth and development. In this work, we utilized our existing 1-DOF knee exoskeleton mechanism (Chen et al., Reference Chen, Hochstein, Kim, Tucker, Hammel, Damiano and Bulea2021) to experimentally evaluate and compare the novel GBAC, Slotine–Li, and PID controllers during a trajectory tracking task. The experimental evaluation provides a focused application of controller performance in the presence of unmodeled system dynamics induced by a spring representing a possible change in user muscle tone or spasticity as well as the addition of limb mass.

4.1. Simulation study

4.1.1. 2-DOF exoskeleton system

We first focus on a lower-limb exoskeleton with 2-DOF at the hip and knee joint in the sagittal plane, where each has flexible joint capability (Figure 1). The inertia, Coriolis matrix, and gravitational vector for the exoskeleton’s equations of motion (equation (3)) are defined as

(23) $$ M(q)=\left[\begin{array}{cc}{\theta}_1+2{\theta}_2\cos \left({q}_2\right)& {\theta}_3+{\theta}_2\cos \left({q}_2\right)\\ {}{\theta}_3+{\theta}_2\cos \left({q}_2\right)& {\theta}_3\end{array}\right], $$

(24) $$ C\left(q,\dot{q}\right)=\left[\begin{array}{cc}-{\theta}_2\sin \left({q}_2\right){\dot{q}}_2& -{\theta}_2\sin \left({q}_2\right)\left({\dot{q}}_1+{\dot{q}}_2\right)\\ {}{\theta}_2\sin \left({q}_2\right){\dot{q}}_1& 0\end{array}\right], $$

(25) $$ G(q)=\left[\begin{array}{c}{\theta}_4g\cos \left({q}_1+{q}_2\right)+{\theta}_5g\cos \left({q}_1\right)\\ {}{\theta}_4g\cos \left({q}_1+{q}_2\right)\end{array}\right], $$

where $ {\theta}_2={l}_1{l}_2{m}_2 $ , $ {\theta}_3= $ $ {l}_2^2{m}_2 $ , $ {\theta}_4={\theta}_3/{l}_2 $ , $ {\theta}_1={l}_2^2{m}_2+{l}_1^2\left({m}_1+{m}_2\right) $ , $ {\theta}_5=\left({\theta}_1-{\theta}_3\right)/{l}_1 $ , and the masses of the exoskeleton links are chosen as $ {m}_1= $  2 kg, $ {m}_2 $  = 0.5 kg, $ {l}_1 $  = 0.35 m, and $ {l}_2 $  = 0.25 m, which are based on our existing exoskeleton hardware. The desired trajectory is set as $ {q}_d=\left[0.5\sin (t),\mathrm{0.5}\sin (2t)+0.6\cos (t)\right] $ . The simulation study implements the proposed GBAC control update law (equation (22)) and compares its performance with the well-known Slotine–Li adaptive controller (Huang and Chien, Reference Huang and Chien2010; Abdelhady and Simon, Reference Abdelhady and Simon2020) since it has a stable adaptation based on Lyapunov stability. All the tuning procedures for the Slotine–Li controller were performed as described in Abdelhady and Simon (Reference Abdelhady and Simon2020). The PID controller was selected as a baseline controller. The following assumptions are made during the simulation:

1. 1. The desired joint trajectory $ {q}_d(t)\in {\mathrm{\mathbb{C}}}^n $ is continuous, bounded differentiable up to the fourth order.

2. 2. $ q,\dot{q} $ and $ {q}_m,{\dot{q}}_m $ are measurable during $ t\in \left[0,\infty \right) $ .

3. 3. All unmodeled nonlinear dynamics, such as backlash and friction, are considered in the velocity motion profile.

4. 4. The hip and knee trajectories imply the existence of the concept of persistent excitation.

The simulation study considers two cases. In the first, the update laws for GBAC and Slotine–Li controllers were calculated with mass and link lengths near to the model values. The second case considers controller behavior in a high uncertainty situation, that is, where all the exoskeleton linkage weights and lengths are 100% above the nominal values of the mechanical model. Because the PID controller does not have an adaptation law, the PID gains were first tuned using link lengths and weights 5% above the nominal values of the original mathematical model. PID performance was then tested twice, at the model values and with model parameters 100% above the original model. The PID gains were tuned using the MATLAB control system toolbox and optimization toolbox. Hip and knee PID controller gains are [33.6, 19.8, and 10.7] and [40, 12, 27] for P, I, and D gains, respectively. The robust adaptive BS controller was implemented as in Narayan et al. (Reference Narayan, Abbas and Dwivedy2023). The Slotine–Li controller adaptation law derivation was described in Abdelhady and Simon (Reference Abdelhady and Simon2020).

To simulate the proposed adaptive control scheme, we identified five Gaussian regressor nodes for each joint (equation (13)). The initial value of each node is 0, and the initial values of $ \lambda $ and $ \eta $ are selected using trial and error to be [10 10] and 100, respectively. Next, the MATLAB single-step solver (ode23) was utilized to solve equation (22). All simulations were carried out with a sampling time of 10 ms. The root mean square (RMS) error from the reference trajectory was computed for the first 2 s of the simulation to evaluate the controller’s performance and convergence.

When the model parameters were close to their expected levels, the tracking performance of all controllers was similarly effective, with convergence in <0.6 s and RMS error <0.4 rad (Figure 2). Notably, GBAC had the fastest convergence (<0.2 s). The performance contrast between controllers was more apparent at the knee. GBAC and BS converged faster than the Slotine–Li controller, with BS converging slightly faster and with a slightly lower RMS (0.66 rad) compared with GBAC (0.76 rad), Slotine–Li (0.86 rad), and PID (1.1 rad). This improved performance can be attributed to the GBAC and BS adaptation mechanisms, which are based on regression and robustness, rather than the Lyapunov-based stability adaptation deployed in Slotine–Li. The PID controller provides relatively good tracking at the hip, whereas at the knee PID controller error was the highest due to the uncontrolled coupling between the two joints.

Figure 2.

(a) Controller performance with model parameters for link mass and length 5% above the mathematical model at the hip (top) and the knee (bottom) indicated by joint angles and error from reference trajectory for Gaussian-based adaptive control ( $ {q}_{\mathrm{GBAC}} $ ), Slotine–Li ( $ {q}_{\mathrm{sl}} $ ), BS ( $ {q}_{\mathrm{BS}} $ ), and PID ( $ {q}_{\mathrm{pid}} $ ), respectively. The insets show controller convergence in the first two cycles. (b) Controller performance presented in the same format as (a) but with model parameters for link mass and length 100% above those in the mathematical model.

As expected, the PID controller had the worst tracking performance due to high uncertainty from the extra mass and length, and the lack of adaptability to handle the coupling effect (Figure 2b). At the hip, GBAC, BS, and Slotine–Li converge to less than 0.5-rad RMS error; however, GBAC converged faster than Slotine–Li and BS. At the knee, Slotine–Li shows poorer tacking performance (RMS = 0.47 rad) compared with BS (0.41 rad) and GBAC (0.35 rad), which performed the best. GBAC also converged slightly faster than BS in this case.

4.1.2. 1-DOF exoskeleton system

To evaluate the performance of controllers under a wider range of disturbances and uncertainties and facilitate a comparison with our existing exoskeleton utilized in the experimental study, a 1-DOF simulation was completed. The 1-DOF exoskeleton system is modeled as a single pendulum with a torsional spring at the joint, viscous friction, and an external disturbance force $ F(t) $ . The system dynamics are governed by the following equation of motion:

(26) $$ {mL}^2\ddot{q}+b\dot{q}+{k}_tq+ mgL\sin (q)=\tau (t)+F(t)+\Delta (t), $$

where $ q $ is the angular displacement of the exoskeleton, $ \dot{q} $ and $ \overset{"}{q} $ are the angular velocity and acceleration, respectively, $ m $ is the mass of the exoskeleton’s pendulum, $ L $ is the length of the pendulum, $ b $ is the viscous damping coefficient, $ {k}_t $ is the torsional spring constant, $ g $ is the acceleration due to gravity, $ \tau (t) $ is the control input torque applied at the joint, $ F(t) $ is the external disturbance force, and $ \Delta (t) $ represents the uncertainty in the system parameters. The system is initialized at $ {q}_0=\pi /2 $  rad, and the controllers aim to track a reference trajectory defined as

(27) $$ {q}_{d1}(t)=\frac{\pi }{4}\mathit{\sin}(t) $$

while rejecting the disturbance $ F(t) $ and compensating for the uncertainty $ \Delta (t) $ .

We first evaluated the ability of three previously described controllers – GBAC, Slotine–Li, and PID – to handle a 100 N external disturbance force $ F(t) $ applied briefly between 2 and 2.25 s. The disturbance is modeled as

(28) $$ F(t)=100\cdot \left(u\left(t-2\right)-u\left(t-2.25\right)\right), $$

where $ u(t) $ is the unit step function. As shown in Figure 3, the GBAC controller, implemented with five Gaussian nodes, demonstrated superior performance in maintaining trajectory tracking despite the disturbance, compared with the Slotine–Li and PID controllers, with an RMS error of 0.27 rad across the simulation window (5 s). The PID controller, with gains $ {K}_p=100 $ , $ {K}_i=37 $ , and $ {K}_d=12 $ , exhibited a slower response and larger error during the disturbance period, resulting in the largest RMS (0.44 rad). The Slotine–Li controller exhibited performance closer to the GBAC controller, but with a higher RMS error of 0.35 rad.

Figure 3.

Simulation results comparing the performance of controllers in response to a 100 N external disturbance force $ F $ applied between 2 and 2.25 s. The system starts from the initial conditions at $ \pi /2 $  rad. The plots show the simulated reference trajectory $ {q}_{d1} $ and the system’s tracking response under GBAC, Slotine–Li, and PID controllers. The insets show the initial convergence and effect of the applied disturbance and recovery dynamics.

Finally, we examined the behavior of the GBAC under parameter uncertainties in the 1-DOF exoskeleton. The initial condition is set at $ {q}_0=\pi /4 $  rad, with the exoskeleton tasked to follow a sinusoidal reference trajectory. The nominal parameters of the exoskeleton were selected to match operational values for human applications, including the frequency of the sinusoidal trajectory and the physical parameters of the exoskeleton system. The simulation was then completed to evaluate the GBAC tracking performance of the exoskeleton joint angle $ q $ in the presence of a set of unexpected parameter changes.

Figure 4a shows the effect of increasing the spring stiffness by 100% above the nominal value of 25 N/m. GBAC quickly adapts to this disturbance and closely tracks the reference trajectory with an overall RMS error of 0.12 rad across the simulation window (5 s). Figure 4b explores the influence of a 50% increase in pendulum length (nominal value 0.35 m), with a greater impact on system performance than spring stiffness (RMS = 0.17 rad). Figure 4c demonstrates the effect of doubling the damping coefficient on the system’s oscillatory behavior, with the largest impact of the simulated parameter changes (RMS = 0.42 rad). Figure 4d illustrates the relative robustness of GBAC to an unexpected increase in mass (100% of the nominal value of 0.5 kg), resulting in an RMS error of 0.19 rad. We also completed simulations to evaluate the Slotine–Li and PID controllers in the presence of spring stiffness and additional mass, and those results are discussed in comparison with the experimental study below. Collectively, these simulations demonstrate the GBAC controller’s robustness and adaptability to maintain suitable system tracking performance in the presence of unmodeled disturbances and significant parameter uncertainties.

Figure 4.

Simulation of a 1-DOF exoskeleton tracking reference trajectory (red dashed line) with the proposed Gaussian-based adaptive control controller (black line) with the following representative uncertainties. (a) Spring stiffness increased by 100% (nominal: 25 N/m). (b) Pendulum length increased by 50% (nominal: 0.35 m). (c) Damping coefficient increased by 100% (nominal: 0.08 N s/m). (d) Mass increased by 100% (nominal: 0.5 kg).

4.2. Experimental study

Whereas the simulation study evaluated controller performance in 1-DOF and 2-DOF exoskeleton, our experimental evaluation focused on evaluating the adaptive control strategies in our existing 1-DOF knee exoskeleton, which was designed for gait assistance and rehabilitation in children with a limb length of 42–66 cm and a total limb mass of 0.8–3.4 kg who have knee extension deficiency from cerebral palsy and other movement disorders. The mechanical and electrical design details of the exoskeleton have been previously reported (Chen et al., Reference Chen, Hochstein, Kim, Tucker, Hammel, Damiano and Bulea2021) and are summarized briefly here. The knee exoskeleton is a two-link mechanism actuated by a geared BLDC motor (Figure 5). The actuator is a high-precision motor (Maxon 323218) equipped with a two-stage gear reduction. The first stage is a planetary gearbox (Maxon 370782) with an 89:1 reduction ratio. The second stage is a right-angle bevel gearbox with a 3:1 reduction ratio.

Figure 5.

The 1-DOF knee exoskeleton used in the experimental study, including the Raspberry Pi for the implementation of the closed-loop control and the Maxon servo drivers for issuing motor commands.

Hall effect sensors and a 512 counts per turn digital encoder (Maxon MR 128–512) measure the rotor and shaft position (and speed), respectively. A potentiometer is attached to the motor shaft to calibrate the encoder. In this experimental setup, Link 1 has one side of a tension spring (25 N/m spring constant) attached 5 cm proximal to the knee joint, with the other side attached to Link 2, 15 cm distal to the joint. The spring was intended to represent unmodeled dynamics typically encountered during exoskeleton use, that is, the undesired muscle force created in response to exoskeleton assistance, possibly due to spasticity.

A servo control module (Escon 50/8) powered the knee exoskeleton actuator. The module has built-in PD current control capability and an accurate current measurement that implicitly represents the joint torque and power. The mechanical and electrical parameters of the 1-DOF knee exoskeleton and actuator are provided in Table 1.

Table 1.

Electrical and mechanical parameters of the 1-DOF knee exoskeleton

The adaptive control laws were implemented on a Raspberry Pi (model B) single-board computer (Figure 5). The Pi calculates the torque command for the exoskeleton (e.g., using equations (16) and (17) for the GBAC controller), which was then scaled as an analog control signal between 0 and 5 V and sent as an analog input to the Escon module to set the motor torque value. Like the simulation, five Gaussian regressor nodes were selected, with an initial value of 0 and initial values of $ \lambda $ and $ \eta $ set to [10 10] and 75, respectively. The overall sampling time for the closed-loop adaptive controllers was 20 ms, including the measurement and calculation latency between the Escon module and the Raspberry Pi. However, for the PID controller, a higher sampling rate (12 ms) was used because of its less complex calculations. The PID gains were set as [43, 16, and 33] for P, I, and D gains, respectively. The recording time for the knee joint position and velocity on the Raspberry Pi SIM card contributed approximately 5-ms latency for every 500 samples. The reference trajectory was selected to be a sinusoidal signal with a frequency of 1 Hz and an amplitude of 0.785 rad (45 degrees). The tension spring of 25 N/m constant was utilized to form a flexible actuated joint and was set at an initial angular position of 45 degrees with the vertical axis. The joint angular position and velocity were recorded by the motor encoder during the testing session for each controller.

Figure 6a shows the experimental study outcomes. The experimental results showed that the GBAC had an initial RMS error that was similar to the Slotine–Li and PID controllers; however, the GBAC quickly converged to a solution with a lower mean RMS error, eventually reaching 0.2 rad after 4 s. The Slotine–Li controller provided stable tracking but with a slower convergence rate than the GBAC control. The RMS error for the final 1 s was 0.64 rad. Finally, the PID controller showed stable but oscillatory tracking around the reference trajectory, with a consistent RMS error of 0.71 rad. The PID performance was attributed to the uncertainty due to the spring force, for which the PID controller is less effectively able to compensate compared with the adaptive control approaches. These experimental results align with those from the 1-DOF simulations. Collectively, the simulation results showed a lower RMS error than in the experiment (Table 2); however, in both the simulation and the experiment, the GBAC had the lowest RMS error, followed by Slotine–Li and then PID.

Figure 6.

(a) Experimental results from the 1-DOF tracking task with a 25 N/m tension spring displayed as the measured knee angle (red) and the reference angle (blue) for the Gaussian-based adaptive control (top), Slotine–Li (middle), and PID (bottom) controllers. (b) Controller performance presented in the same format as (a) but with a 0.5-kg mass added to the shank center of mass. A low-pass filter with a 20-Hz cutoff frequency was used to smooth measured angular position and velocity.

Table 2.

Comparison of simulation and experimental results

To further validate the GBAC performance in controlling a flexible joint (mechanism plus spring), a second experiment was performed with a 0.5-kg mass added to the shank center of mass. This makes the exerted external force vary between 4.9 N at the horizontal position of the shank and 2.45 N at the end of the range of motion. This additional mass was intended to simulate the weight of a limb that would not be present during initial system tuning.

The tracking performance of all controllers is clearly affected by the presence of the additional mass (Figure 6b). Both the GBAC and Slotine–Li controllers converge under this situation of high uncertainty (variable shank mass) and disturbance (spring flexibility), which constitute a considerable challenge for controllers without adaptation, such as the PID controller. Notably, the PID controller, as shown in Figure 6b, displayed the worst tracking performance with an RMS error value over the 5-s test period of 1.56 rad. On the other hand, GBAC and Slotine–Li were able to adapt and show convergence within 5 s. The Slotine–Li controller shows an oscillatory behavior during its convergence, resulting in an overall tracking RMS error of 0.87 rad. Similar oscillatory behavior was observed with GBAC during the first 2 s; however, GBAC overcomes this, likely due to the adaptation facilitated by the Gaussian nodes, as shown in equation (14). As in the first experiment, the results with additional mass followed the same pattern as the 1-DOF simulation, with RMS error being higher for all controllers in the experimental results (Table 2). Interestingly, the GBAC showed less RMS error than the Slotine–Li in simulation but similar in the experiment. In both cases, the PID controller had the worst tracking performance with the unexpected addition of mass to the flexible joint.