2.1. Dynamic model

We use a two-link leg model to represent swing dynamics

(2.1) $$ D(q)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+{M}_{\mathrm{ev}}\left(q,\dot{q}\right)+G(q)+W=\tau +{\tau}_{\mathrm{ext}}\sigma, $$

where $ q={\left[{q}_h,{q}_k\right]}^T $ represents the hip and knee angles, respectively, with their first and second time derivatives denoted by $ \dot{q},\ddot{q}\in {\mathrm{\mathbb{R}}}^2 $ . $ D,C\in {\mathrm{\mathbb{R}}}^{2\times 2} $ are the inertial and Coriolis matrices, $ {M}_{ev},G,W\in {\mathrm{\mathbb{R}}}^2 $ are the passive viscoelastic, gravitational, and unknown disturbance torques, and $ \tau \in {\mathrm{\mathbb{R}}}^2 $ is the active torque due to motors and FES. The ground reaction torque is represented by $ {\tau}_{ext}\in {\mathrm{\mathbb{R}}}^2 $ and is multiplied by a switching signal $ \sigma =\left\{0,1\right\} $ , where $ \sigma =0 $ during the swing phase and $ \sigma =1 $ during the stance phase. The joint-level torques $ {\tau}_i $ , $ i=\left[h,k\right] $ , may be broken down as

(2.2) $$ {\tau}_i={T}_{m,i}-{T}_{s,i}^e+{T}_{s,i}^f, $$

where $ {T}_{m,i}\in \mathrm{\mathbb{R}} $ is the torque due to the motor at joint $ i $ and $ {T}_{s,i}^e,{T}_{s,i}^f\in \mathrm{\mathbb{R}} $ are the torques due to FES activation of the extensor and flexor muscles, respectively. For $ j=\left[e,f\right] $ , the FES-elicited torques are calculated as

$$ {T}_{s,i}^j={\mu}_i^j{\eta}_i^j{u}_{a,i,{\tau}_i^j}^j $$

for MEI $ {\mu}_i^j\in \left[{\varsigma}_i^j,1\right] $ , unknown bounded nonlinear function $ {\eta}_i^j\left(q,\dot{q}\right)\in {\mathrm{\mathbb{R}}}^{+} $ , and delayed control input $ {u}_{a,i,{\tau}_i^j}^j={u}_{a,i,}^j\left(t-{\tau}_i^j\right) $ . The constant electromechanical delay is represented as $ {\tau}_i^j\in {\mathrm{\mathbb{R}}}^{+} $ , and the input is normalized so that $ {u}_{a,i,{\tau}_i^j}^j\in \left[0,1\right] $ .

The MEI quantifies the degree of muscle fatigue and is subject to its own dynamics

(2.3) $$ {\dot{\mu}}_i^j={w}_{f_i}^j\left({\varsigma}_i^j-{\mu}_i^j\right){u}_{a,i,{\tau}_i}^j+{w}_{r_i}^j\left(1-{\mu}_i^j\right)\left(1-{u}_{a,i,{\tau}_i}^j\right), $$

where $ {w}_{f_i}^j,{w}_{r_i}^j\in {\mathrm{\mathbb{R}}}^{+} $ are fatigue and recovery time constants. A fully rested muscle will have an MEI of 1, whereas a fully fatigued muscle will have an MEI of $ {\varsigma}_i^j\in \left(0,1\right) $ .

The motor torque in (2.2) is given by

$$ {T}_{m,i}={K}_{m,i}{u}_{m,i} $$

for motor constant $ {K}_{m,i}\in {\mathrm{\mathbb{R}}}^{+} $ and normalized input $ {u}_{m,i}\in \left[-1,1\right] $ .

2.2. Switching controller

The control objective is to drive the joint angles $ q $ toward a desired trajectory $ {q}_d(t)\in {\mathrm{\mathbb{R}}}^2 $ using two modes. These modes are driven by a switching signal $ \xi \in \mathrm{\mathbb{R}}. $ When the hybrid mode (Mode H) is active, $ \xi =1 $ , whereas when the exoskeleton-only mode (Mode E) is active, $ \xi =-1 $ . A high-level diagram of the switching control scheme is provided in Figure 1. The switching controller alternates between modes based on muscle fatigue feedback. The controller begins in Mode H and remains in Mode H, with all MEIs initialized to 1. When at least one MEI falls below a predetermined threshold $ \underline{\mu} $ , the controller switches to Mode E. Once all muscles have recovered such that their MEIs are above another predetermined value $ \overline{\mu}>\underline{\mu} $ , the controller switches back to Mode H, and so forth. The switching logic is represented as follows:

$$ {\xi}_{t_k+1}=\left(\begin{array}{cc}1,& \left\{\begin{array}{cc}{\mu}_i^j\ge \underline{\mu},& {\xi}_{t_k}=1,\\ {}{\mu}_i^j\ge \overline{\mu},& {\xi}_{t_k}=-1\end{array}\right.\\ {}-1& \left\{\begin{array}{cc}{\mu}_i^j\le \underline{\mu},& {\xi}_{t_k}=1,\\ {}{\mu}_i^j\le \overline{\mu},& {\xi}_{t_k}=-1\end{array}\right.\end{array}\right.. $$

Figure 1.

High-level diagram of the switching control scheme. (a) The exoskeleton used in the experiments. (b) The switching control scheme. The controller alternates between hybrid (Mode H) and exoskeleton-only (Mode E) control based on real-time US-derived MEI values. There is no FES-only mode in the switching controller.

To describe the controller terms in the two modes, first, we introduce a proportional error $ e\in {\mathrm{\mathbb{R}}}^2 $ as

$$ {e}_i={q}_{d,i}-{q}_i $$

and an auxiliary error $ r\in {\mathrm{\mathbb{R}}}^2 $ as

$$ r=\dot{e}+\alpha e-\frac{\xi +1}{2}\beta {e}_c $$

for constants $ \alpha, \beta \in {\mathrm{\mathbb{R}}}^{+} $ , switching signal $ \xi =\left\{-1,1\right\} $ , and delay compensation term $ {e}_c\in {\mathrm{\mathbb{R}}}^2 $ . The delay compensation vector is calculated as

$$ {e}_c={e}_{c,i}^f-{e}_{c,i}^e $$

for the vectors $ {e}_{c,i}^j\in {\mathrm{\mathbb{R}}}^2 $ given by

$$ {e}_{c,i}^j={\int}_{t-{\tau}_i^j}^t{u}_{s,i}^j\left(\theta \right)\hskip0.1em d\theta . $$

The FES-elicited torques are given as

(2.4) $$ {T}_{s,i}^j=\frac{1+{\xi}_{\tau_i^j}}{2}{\mu}_i^j{\eta}_i^j{K}_{a,i}^j{u}_{s,i,{\tau}_i^j}^j $$

for constant gain $ {K}_{a,i}^j\in \mathrm{\mathbb{R}} $ . The motor torque is given by

(2.5) $$ {T}_{m_H,i}=\frac{1+\xi }{2}{K}_{m,i}{u}_{m_H,i} $$

for Mode H, where $ {u}_{m_H} $ is the motor input under Mode H, and by

(2.6) $$ {T}_{m_M,i}=\frac{1-\xi }{2}{K}_{m,i}{u}_{m_E,i} $$

for Mode E, where $ {u}_{m_E} $ is the motor input under Mode E. Combining (2.1) and (2.4)–(2.6) yields

$$ {\displaystyle \begin{array}{c}D\ddot{q}+C\dot{q}+{M}_{ev}+G+W={\left(\frac{1+{\xi}_{\tau_i^f}}{2}{\mu}_i^f{\eta}_i^f{K}_{a,i}^f{u}_{s,i,{\tau}_i^f}^f\right)}_2\\ {}-{\left(\frac{1+{\xi}_{\tau_i^e}}{2}{\mu}_i^e{\eta}_i^e{K}_{a,i}^e{u}_{s,i,{\tau}_i^e}^e\right)}_2+{\left(\frac{1+\xi }{2}{K}_{m,i}{u}_{m_H,i}\right)}_2+{\left(\frac{1-\xi }{2}{K}_{m,i}{u}_{m_E,i}\right)}_2.\end{array}} $$

2.2.1. Hybrid exoskeleton mode (H)

Mode H is governed by the feedback law

(2.7) $$ \left[\begin{array}{c}{u}_s^f\\ {}{u}_s^e\\ {}{u}_{m_H}\end{array}\right]={K}_u\left[\begin{array}{c}{Q}_1^f{Q}_2{Q}_3\\ {}{Q}_1^e{Q}_2{Q}_3\\ {}{Q}_4\end{array}\right]r, $$

where $ {K}_u\in {\mathrm{\mathbb{R}}}^{+} $ and $ {Q}_1^f,{Q}_1^e,{Q}_2,{Q}_3,{Q}_4\in {\mathrm{\mathbb{R}}}^{2\times 2} $ are given as

$$ {\displaystyle \begin{array}{c}{Q}_1^f=\operatorname{diag}\left(\frac{1+\operatorname{sgn}\left({r}_1\right)}{2},\frac{1+\operatorname{sgn}\left({r}_2\right)}{2}\right),\\ {}{Q}_1^e=\operatorname{diag}\left(\frac{1-\operatorname{sgn}\left({r}_1\right)}{2},\frac{1-\operatorname{sgn}\left({r}_2\right)}{2}\right),\\ {}{Q}_2=\operatorname{diag}\left({\rho}_{s,1}{\rho}_{s,2}\right),\\ {}{Q}_3=\operatorname{diag}\left(\operatorname{sgn}\left({r}_1\right),\operatorname{sgn}\left({r}_2\right)\right)\\ {}{Q}_4={\left({\rho}_{m,i,j}\right)}_{2\times 2}\end{array}} $$

for constant control gains $ {\rho}_{s,i}\in {\mathrm{\mathbb{R}}}^{+} $ . The electric motor gains are represented by $ {\rho}_{m, ij}\in \mathrm{\mathbb{R}} $ . The signum functions in $ {Q}_1,{Q}_3 $ account for the fact that the FES-elicited muscle torques are monoarticular and act in opposite directions. (Note that the FES-only controller utilized in treadmill walking is the same as (2.7) but without $ {u}_{m_H} $ .)

2.2.2. Exoskeleton-only mode (E)

Mode E is given by

(2.8) $$ {u}_{m_E}={K}_m^{-1}\left(\frac{r}{\left\Vert r\right\Vert +{r}_c}\left(\delta {\varPhi}^{\prime}\left(\left\Vert {y}_{er}\right\Vert \right)\left\Vert {y}_{er}\right\Vert +{\varPsi}^{\prime}\right)+{K}_vr\right) $$

for constant gain $ {K}_v\in {\mathrm{\mathbb{R}}}^{+} $ , small constant $ {r}_c\in {\mathrm{\mathbb{R}}}^{+} $ , and error vector $ {y}_{er}={\left[{e}^T,{r}^T\right]}^T\in {\mathrm{\mathbb{R}}}^4 $ . The guessing functions $ \delta {\varPhi}^{\prime } $ , which is positive globally invertible and nondecreasing, and $ {\varPsi}^{\prime}\in {\mathrm{\mathbb{R}}}^{+} $ are prescribed as

(2.9) $$ \delta {\varPhi}^{\prime }=0.001\left\Vert {y}_{er}\right\Vert $$

(2.10) $$ {\varPsi}^{\prime }=1+0.01\left|{\varPhi}_d\right|, $$

where $ {\varPhi}_d(t)\in {\mathrm{\mathbb{R}}}^2 $ is the desired torque obtained from (2.1) given $ {q}_d(t),{\dot{q}}_d(t),{\ddot{q}}_d(t) $ .

2.2.2.1. Remark

Details of control development as well as conditions for stability can be found in Sheng et al. (Reference Sheng, Sun, Molazadeh and Sharma2021a). While our previous work establishes rigorous stability criteria, deriving gain conditions that strictly satisfy these criteria under realistic physiological variability would require substantial additional analysis. Importantly, these criteria are sufficient – but not necessary – for stability, which allows the present study to evaluate the fatigue-compensation abilities of the controller using heuristically tuned gains that achieve safe and satisfactory performance. We selected functions similar to those in Sheng et al. (Reference Sheng, Sun, Molazadeh and Sharma2021a). The magnitudes were lower than those in Sheng et al. (Reference Sheng, Sun, Molazadeh and Sharma2021a) because, during initial testing, $ \delta {\varPhi}^{\prime}\left(\left\Vert {y}_{er}\right\Vert \right)\left\Vert {y}_{er}\right\Vert +{\varPsi}^{\prime } $ was large enough that it dominated $ {u}_{m_E} $ as defined in (2.8). Reducing the magnitudes of these functions allowed the $ {K}_vr $ term to influence controller behavior.

2.3. Sampled-data observer

A sampled-data observer as described in Iyer et al. (Reference Iyer, Sun, Lambeth, Singh, Cleveland and Sharma2024) was used to update the model-based MEI estimate in real time. Let the model-based MEI estimate of muscle $ i,j $ be $ {\hat{\mu}}_i^j $ , and let the US-derived MEI estimate at sample time $ {t}_k $ be $ {\mu}_i^j\left({t}_k\right) $ . The sampled-data structure combines the two estimates as

(2.11) $$ {\dot{\hat{\mu}}}_i^j={w}_{f_i}^j\left({\varsigma}_i^j-{\hat{\mu}}_i^j\right){u}_{s,i,{\tau}_i^j}^j+{w}_{r_i}^j\left(1-{\hat{\mu}}_i^j\right)\left(1-{u}_{s,i,{\tau}_i^j}^j\right)+\gamma {\tilde{\mu}}_i^j\left({t}_k\right), $$

where $ \gamma \in {\mathrm{\mathbb{R}}}^{+} $ is a user-defined constant. A larger $ \gamma $ places greater emphasis on the US-derived MEI. The error term $ {\tilde{\mu}}_i^j\left({t}_k\right) $ is given by

$$ {\tilde{\mu}}_i^j\left({t}_k\right)={\hat{\mu}}_i^j\left({t}_k\right)-{\mu}_i^j\left({t}_k\right). $$

This value is updated only when a new US-derived measurement is available. Conditions for asymptotic convergence of $ {\tilde{\mu}}^i $ to 0 may be found in Iyer et al. (Reference Iyer, Sun, Lambeth, Singh, Cleveland and Sharma2024).

We presented the algorithm for calculating US-derived axial strain in Sheng et al. (2020), where we established a correlation between FES-induced fatigue of the quadriceps and axial strain during isometric contractions. The basis of this algorithm is finding the displacement between successive frames at each 2D coordinate using an adaptive speckle-tracking algorithm (Lubinski et al., 1999). For stronger contractions, greater muscle tissue motion is produced, and therefore the average strain value across a 2D region of the muscle is larger. As muscle force and tissue motion decay with the onset of fatigue, strain also decays. We therefore consider a fully rested muscle to have an MEI of $ \mu =1 $ , whereas a muscle with an MEI of $ 0 $ is fatigued to the extent that it no longer responds to FES.

3.1. Stimulation mapping

Prior to experiments, each participant’s threshold and saturation currents, $ {I}_t\in {\mathrm{\mathbb{R}}}^{+} $ and $ {I}_s\in {\mathrm{\mathbb{R}}}^{+} $ , respectively, were identified using the methods in Kirsch et al. (Reference Kirsch, Alibeji and Sharma2017). For both the stance and swing phases, the current supplied to the quadriceps, $ {I}_Q\in {\mathrm{\mathbb{R}}}^{+} $ , and to the hamstrings, $ {I}_H\in {\mathrm{\mathbb{R}}}^{+} $ , were calculated as follows:

(3.1) $$ {\displaystyle \begin{array}{l}{I}_Q=\left\{\begin{array}{ll}{I}_{s_Q},& {u}_s^e\ge 1,\\ {}{u}_s^e\left({I}_{t_Q}-{I}_{s_Q}\right)+{I}_{s_Q},& 1>{u}_s^e>0,\\ {}0,& {u}_s^e\le 0,\end{array}\right.\\ {}{I}_H=\left\{\begin{array}{ll}{I}_{s_H},& {u}_s^f\ge 1,\\ {}{u}_s^f\left({I}_{t_H}-{I}_{s_H}\right)+{I}_{s_H},& 1>{u}_s^f>0,\\ {}0,& {u}_s^f\le 0.\end{array}\right.\end{array}} $$

For cases where $ {u}_s^e $ , $ {u}_s^f\le 0 $ , the decision was made to map $ {u}_s^f $ , $ {u}_s^e $ to 0 mA rather than to $ {I}_t $ to avoid agonist–antagonist coactivation and constant muscle stimulation. Figure 3 illustrates how (3.1) maps control input to current.

Figure 3.

The top row depicts the hamstrings and quadriceps control inputs from (2.7) ( $ {u}_s^f $ and $ {u}_s^e $ , respectively) over one gait cycle of FES-only control. The middle row shows how these inputs are mapped to hamstrings current $ {I}_H $ and quadriceps current $ {I}_Q $ . The bottom row shows the actual and desired knee angles during the gait cycle as well as the knee tracking error, defined as $ {q}_d-q $ .

Figure 3. Long description

The figure consists of six line graphs, all sharing a horizontal X-axis of Time in seconds from 0 to 8.

Top-Left Panel: Normalized Hamstrings Input u sub s super f. The Y-axis ranges from 0 to 1.5. The data shows two distinct peaks: a smaller jagged peak around 4 seconds and a sharp, narrow peak reaching 1.5 at approximately 5.5 seconds.

Top-Right Panel: Normalized Quadriceps Input u sub s super e. The Y-axis ranges from 0 to 2. The data shows high activity at the start, dropping near 0 at 3.5 seconds, then rising with multiple jagged peaks between 5 and 8 seconds.

Middle-Left Panel: Hamstrings F E S current I sub H in m A. The Y-axis ranges from 0 to 60. The trend mirrors the top-left graph, showing current spikes corresponding to the normalized inputs, peaking at 60 m A.

Middle-Right Panel: Quadriceps F E S current I sub Q in m A. The Y-axis ranges from 0 to 60. The trend mirrors the top-right graph, with current levels fluctuating and saturating at 60 m A during the active phases.

Bottom-Left Panel: Knee Angle in degrees. The Y-axis ranges from -10 to 60. It compares ‘Actual’ (solid blue line) and ‘Desired’ (dashed red line). The desired path shows a trapezoidal pulse between 4.5 and 6.5 seconds. The actual path follows the general shape but exhibits higher peaks and oscillations.

Bottom-Right Panel: Knee Tracking Error in degrees. The Y-axis ranges from -40 to 20. The error fluctuates significantly, starting at -40, rising toward 0, and showing a sharp positive spike to 20 degrees at 5.5 seconds before stabilizing around -20 degrees.

3.2. Gait segmentation and desired trajectory generation

The finite state machine, which is depicted in Figure 4 and was previously presented in Iyer et al. (Reference Iyer, Sun, Lambeth, Singh, Cleveland and Sharma2024), is time-based, not event-based. For participants with disabilities, the 2-s initialization phase is followed by a 20-s sit-to-stand phase. No motor assistance is provided during the initialization phase. A 10-s rest phase occurs before the onset of the right half-step. Note that the first step is a half-step because walking starts with the feet together. In participants with no disabilities, the initialization phase is immediately followed by the right half-step phase. The right half-step, as well as all subsequent full step phases, is 3.5 s in duration. The right half-step is followed by a 0.75-s rest/double support phase (DSP) wherein all desired trajectories are held constant. Next is the first full left step, and the finite state machine continues cycling between full steps and DSP phases until the experiment is finished. (Note that, for participant A3, the roles of the left and right leg were switched because data were collected from the right leg, not the left. This was because this participant’s right leg was significantly more responsive to FES than the left leg.) The desired trajectories themselves are created using third-order polynomial time scaling (Lynch and Park, Reference Lynch and Park2024), which requires only the duration and initial and final desired angles of each subphase.

Figure 4.

(a) The finite state machine governing both treadmill and overground walking. DSP = double support phase. $ st= $ state time, or the time since the last state change. (b) Desired trajectories with gait phase.

Figure 4. Long description

Panel a is a flowchart of a finite state machine. It begins at Initialize, moving right to Sit-to-Stand when s t is greater than 2 s, then right to Rest when s t is greater than 20 s, and right to Right Half-Step when s t is greater than 10 s. From Right Half-Step, the path splits: one arrow leads down to Right D S P when s t is greater than 3.5 s, and another leads left to Right Step when s t is greater than 3.5 s. The main gait cycle forms a loop: Right D S P leads left to Left Step when s t is greater than 0.75 s; Left Step leads up to Left D S P when s t is greater than 3.5 s; Left D S P leads right to Right Step when s t is greater than 0.75 s; and Right Step leads down back to Right D S P when s t is greater than 3.5 s.

Panel b contains two line graphs with Time in seconds on the x-axis.

1. Desired Hip Trajectory: The y-axis is Hip Angle in degrees. The curve rises to 50 degrees during the Left Swing Phase (purple background), stays flat through the Left D S P (gray), then drops to 0 during the Right Swing Phase (red) and remains at 0 through the Right D S P (green).

2. Desired Knee Trajectory: The y-axis is Knee Angle in degrees. The curve rises to 50 degrees and falls back to 0 during the Left Swing Phase. It remains at 0 through the Left D S P. During the Right Swing Phase, it shows a small peak of approximately 15 degrees before returning to 0 for the Right D S P.

3.3. Treadmill walking with three control cases

This study was approved by the Institutional Review Board of North Carolina State University (IRB Approval # 20553). Three participants with SCI and three participants with no disabilities walked on a split-belt treadmill (Bertec, OH, USA) while wearing a powered lower-limb exoskeleton (INDEGO, Ekso Bionics, CA, USA) with bilateral hip and knee motors. Participant demographics are reported in Table 1.

Table 1.

Participant demographics

Table 1. Long description

The table consists of five columns: Subject, Age, Gender, Injury level, and Leg.

* Subject A 1: 55, M, T 10, incomplete, L.

* Subject A 2: 62, M, T 7, complete, L.

* Subject A 3: 47, M, T 12, incomplete, R.

* Subject B 1: 21, F, N / A, L.

* Subject B 2: 35, M, N / A, L.

* Subject B 3: 30, M, N / A, L.

Participants used parallel bars for balance, and a physical therapist provided additional stabilization for the SCI group. The Indego possesses built-in FES ports, which delivered stimulation to the quadriceps and hamstrings. Because FES was not applied to the hip muscles, the control laws from Section 2 were applied only to the knee joint, with the hip joint controlled simply as $ {u}_{m,1}={k}_hr $ for constant $ {k}_h\in {\mathrm{\mathbb{R}}}^{+} $ . Furthermore, $ {\tau}_{ext} $ was not measured and thus neglected during the stance phase. The exoskeleton motors and FES were controlled by Simulink Desktop Real-Time using MATLAB (R2020b, MathWorks, MA, USA) with a control frequency of 100 Hz. Stimulation was administered at a pulsewidth of 300 $ \mu $ s and a frequency of 40 Hz. Subject-specific saturation and threshold currents were used, while generalized parameters for $ D $ , $ G $ , and $ {M}_{ev} $ were used to calculate $ {\varPhi}_d $ in (2.8).

Each participant performed at least one hybrid exoskeleton trial and one FES-only trial, with one SCI participant completing multiple trials of each case. One SCI participant and all participants with no disabilities also walked with exoskeleton-only control. Participants walked for 6 min or the maximum amount of time which they could consistently walk, whichever was less. See the Appendix for details regarding the trial orders (Table A2) and for the durations each participant walked (Table A3).

Prior to walking, a linear US transducer (L7.0SC Prodigy Probe, S-Sharp, Taiwan) was secured along the quadriceps over the rectus femoris and vastus intermedius. Visual inspection using live B-mode imaging was used to verify that the location was suitable, that is, that a visible muscle contraction occurred in response to a 2-s FES pulse. For all US imaging, the participant remained seated to ensure an isometric contraction. US images were collected using a Prodigy Research Platform with an NVIDIA Titan V GPU (NVIDIA, CA, USA). Prior to each trial, five FES-elicited contractions were imaged, each for 0.5 s in duration. The same US imaging procedure was conducted immediately concluding each trial. To allow for muscle recovery, a minimum 10-min rest period was provided between walking trials if more than one trial was performed in the same day.

The controllers were tested only on the left knee (this was with the exception of participant A3, for whom the controllers were applied on the right knee due to the left quadriceps not eliciting a strong extension torque in response to FES). The hip motors and contralateral knee motor were controlled using position-based PID control, and a finite state machine governed the desired joint trajectories.

For each trial, root-mean-square (RMS) values of the FES current amplitudes were calculated from the onset of the right (left for participant A3) half-step state to the onset of the final right (left) step state. RMS values for participant A1, who completed multiple trials of each control case, were averaged across all trials of that respective case.

For each contraction in each pre-walking and post-walking trial, the peak strain value in the contraction was identified. All five pre-walking and all five post-walking strain peaks were normalized with respect to the maximum strain peak across both datasets. The average of the normalized pre-walking peaks was taken and considered the mean strain value for the pre-walking dataset, and likewise for the post-walking dataset. The difference between the pre-walking and post-walking mean peak normalized strain values was then determined and considered the change in strain for that trial and participant.

3.4. Overground walking with switching controller

Two participants with SCI and three participants with no disabilities completed overground walking with the switching controller. A participant with SCI can be seen walking in Figure 5. Participants walked for 6 min or the maximum duration they could comfortably walk, whichever was less. The hamstrings MEI estimate was purely model-based using (2.3) with subject-specific parameters identified prior to experiments following the method in Bao et al. (Reference Bao, Sheng, Dicianno and Sharma2021), originally adapted in real-time control from Riener et al. (Reference Riener, Quintern and Schmidt1996). Values of the muscle parameters are reported in the Table A1, and a numerical sensitivity analysis of switching times with parameter perturbations is provided in Figure A1. For the quadriceps, the sampled-data observer in (2.11) was used.

Figure 5.

Exoskeleton-assisted walking frames. A participant completes a step with exoskeleton assistance. (a) The participant starts in double support (left foot forward), (b) initiates right toe off, (c) enters the right swing phase, and (d) ends in double support (right foot forward).

Figure 5. Long description

A sequence of four photos labeled a through d from left to right.

* Panel a. The participant stands with both feet on the ground. The left foot is positioned forward while the right foot is behind. The participant grips the handles of a silver metal walker. The exoskeleton is visible as black mechanical bracing around the waist, thighs, and lower legs.

* Panel b. The participant shifts weight forward. The heel of the right foot is lifted off the ground while the toes remain in contact.

* Panel c. The right leg is lifted entirely off the ground and moves forward in a swing motion. The left leg remains planted for support.

* Panel d. The right foot makes contact with the ground in front of the left foot. Both feet are now on the ground again, completing the step with the right foot forward.

Prior to walking, five isometric FES-induced contractions of the quadriceps were imaged. The mean strain value was considered to correspond to an MEI value of 1, which was used to normalize US-derived strain values during walking so that MEI remained between 0 and 1. The quadriceps were imaged during the stance phase once every four gait cycles, and, once calculated, the computed US-derived MEI was sent to the observer.