The development in this work is based on the hybrid systems framework described in Goebel et al. (2012). A hybrid system H = C , F , D , G with state x ∈ X ⊂ R n is modeled by H : x ̇ ∈ F x x ∈ C x + ∈ G x x ∈ D . (1)

When the state is in the flow set C ⊂ R n , it is allowed to evolve continuously according to the set-valued flow map F : R n ⇉ R n . When the state is in the jump set D ⊂ R n , it is allowed to change discretely according to the jump map G : R n ⇉ R n . When x ∈ C ∩ D, either behavior is possible. The notion of a solution to H is defined precisely in Goebel et al. (2012), Def. 2.6. Briefly, a solution to H is a function t , j ↦ ϕ t , j defined on a hybrid time domain dom ϕ ⊂ R ≥ 0 × N and is parameterized by the ordinary time variable t ∈ R ≥ 0 and the discrete jump variable j ∈ N . The set-valued mappings F and G map points in R n to subsets of R n so that, for example, the inclusion x + ∈ G x represents the fact that if a trajectory jumps from the state ϕ t , j , then its state ϕ t , j + 1 at the next discrete time instant is a point in the set G ϕ t , j .

Remark 1. Previous results (cf. Rouse et al. (2020); Bellman et al. (2017); Rouse et al. (2021)) have analyzed the dynamic model in the subsequent section using switched-systems tools. The decision to use a hybrid model here was motivated by the fact that forward invariance via barrier functions is not well characterized for switched systems, nor are many results available regarding the stability of noncompact sets. Hybrid systems can model broad classes of switched systems Goebel et al. (2012, Section 2.4).

Analogous to Eq. 1, one can also consider hybrid systems with inputs Sanfelice (2013). We use such a system to describe the control design but present our stability analysis in terms of a closed-loop system with the form in Eq. 1. The open-loop cycle-rider system is modeled as a continuous-time system H u = C u , F u . Subsequently, discrete dynamics will be introduced due to the design of the controller. Adapting the model from our previous work in Bellman et al. (2017) and Isaly et al. (2020), the cycle’s Euler-Lagrange dynamics are modeled using the flow map z ̇ ∈ z 2 M − 1 z 1 τ u z , u − τ F z ≜ F u z , u , (2) and flow set C u ≜ R 2 × U . In (2), the state is z ∈ R 2 , where z₁ denotes the cycle’s measurable crank angle, and z₂ is the calculable angular velocity (equivalently, the rider’s cadence). The system has control inputs ¹ u ≜ u e , u M , where u e ∈ R is the current input to the cycle’s electric motor, and u M ∈ R 6 is a vector of the electrical stimulation intensity inputs u m ∈ R , for each muscle m ∈ M ≜ L Q , L G , L H , R Q , R G , R H . The elements of M indicate the quadriceps femoris (Q), gluteal (G), and hamstring (H) muscle groups for the left and right legs, respectively. The control inputs take values in the set U ≜ R × U M , where U M ≜ 0 , u ̄ 6 ⊂ R 6 indicates that the muscle control inputs are bounded by the constant u ̄ > 0 for the rider’s safety and comfort. The continuously differentiable function M : R → R > 0 denotes the inertial forces from the cycle and rider’s legs. The set-valued mapping τ F : R 2 ⇉ R defines the dynamics of the system as τ F z ≜ τ b z 2 + V p z z 2 + G z 1 + P z + T d + T v o l , (3) where τ b : R → R denotes the unknown torque due to viscous damping in the cycle, and V p : R 2 → R , G : R → R , and P : R 2 → R are the unknown centripetal-Coriolis, gravitational, and passive viscoelastic tissue forces, respectively, applied by the combined human-cycle system. The aforementioned functions are continuous according to the dynamic models in Bellman et al. (2017) and Idsø (2002). According to the model in Bellman et al. (2017), the centripetal-Coriolis term is related to the mass and cadence by V p z = 1 2 ∇ M z 1 z 2 . In Eq. 3, T v o l ⊂ R and T d ⊂ R are sets used to model all of the possible values of the rider’s volitional effort and other unknown disturbances, respectively ² . The continuous function τ u : R 2 × U → R describes the torque produced by the control inputs and is defined as τ u z , u ≜ c e u e + τ F E S z , u M , where c_e > 0 is the known electric motor control constant relating input current to output torque. The torque generated from FES inputs to the rider’s muscles τ F E S : R 2 × U M → R is given by τ F E S z , u M ≜ ∑ m ∈ M g m z u m , (4) where the continuous functions g m : R 2 → R denote the uncertain control effectiveness of each muscle. For each m ∈ M , let the closed set Q m ⊂ R denote the portion of the crank cycle when a particular muscle m is stimulated, which is selected based on a minimum threshold for the torque transfer ratio of each muscle group. In particular, there exist kinematic deadzones in the crank cycle where no muscle is able to produce useful torque Cousin et al. (2020).

The following properties of the cycle-rider system in Eq. 2 are derived from a detailed dynamic model, as discussed in Bellman et al. (2017).

Property 1. The inertial term is upper- and lower-bounded as c ̲ I ≤ M z 1 ≤ c ̄ I for all z 1 ∈ R , where c ̲ I , c ̄ I > 0 are known constants.

Property 2. The centripetal-Coriolis parameter is upper-bounded as V p z ≤ c V z 2 for all z ∈ R 2 , where c_V > 0 is a known constant.

Property 3. The torque generated by gravity is upper-bounded as G z 1 ≤ c G for all z 1 ∈ R , where c_G > 0 is a known constant.

Property 4. The torque generated by the rider’s viscoelastic tissues is upper-bounded as P z ≤ c P 1 + c P 2 z 2 for all z ∈ R 2 , where c_P1, c_P2 > 0 are known constants.

Property 5. The torque due to viscous damping is upper-bounded as τ b z 2 ≤ c b z 2 for all z 2 ∈ R , where c_b > 0 is a known constant.

Property 6. The torques generated by system disturbances are bounded so that T d = − c d , c d , where c_d > 0 is a known constant.

Property 7. Due to physical limitations of the rider, the volitional muscle torque is bounded so that T v o l = − c v o l , c v o l , where c_vol > 0 is a known constant.

Property 8. For each m ∈ M , the muscle control effectiveness is upper-bounded so that g m z ≤ c ̄ m for all z ∈ R 2 , where c ̄ m > 0 is a known constant.

Property 9. The set-valued mapping F u : R 2 × U ⇉ R 2 is outer semicontinuous, locally bounded, and convex-valued. These properties follow from continuity of the defining functions and from Properties 6 and 7.

To formalize the control objective, we define the tracking error e as the deviation of the cadence state z₂ from a constant setpoint z_2d > 0, ³ e ≜ z 2 − z 2 d . (5)

The primary control objective is to guarantee a safe and effective therapy by constraining the rider’s cadence to the safe set S ≜ { z ∈ R 2 : e L ≤ e ≤ e H } , where e_L < 0 < e_H are user-defined constants. The FES control inputs attempt to constrain the rider’s cadence to the secondary set S F E S ≜ { z ∈ R 2 : e ≥ e F E S } . Because the FES inputs are only intermittently available and must be less than the comfort threshold u ̄ , the cadence may not remain within S F E S . However, this construction is useful for design purposes. To ensure that FES stimulation is active before torque is added by the electric motor, the design specifies that e_L < e_FES < 0.

The goal is to synthesize, in a systematic way, controllers that render a given set of states uniformly globally asymptotically stable (UGAS) while using the minimum required effort inside the set of interest. ⁴ Combining ideas from CLF theory with recent developments regarding CBFs, this task is accomplished by using a QP to enforce a constraint on the control input that is induced by a candidate barrier function. The following lemma, presented in a more generic form than in our preliminary work in Isaly et al. (2020), gives conditions under which a QP-based control law with a single constraint is feasible and locally Lipschitz continuous. The closed-form solution of the QP in the absence of a nominal controller (but including the case of multiple control inputs) has also been presented in Freeman and Kokotovic (1996, Section 4.2.2) and was developed in detail in Xu et al. (2015), Thm. 8. In those works, the feasibility condition was guaranteed by assuming the existence of a CLF or CBF, respectively. Because the addition of a nominal controller is a minor extension of the available literature, we do not present a proof of the result. Lemma 1 applies to the control laws developed in the subsequent sections.

Lemma 1. Let the functions a : R n → R , b : R n → R , and u n o m : R n → R be locally Lipschitz on R n and satisfy the following feasibility condition: a z = 0 ⇒ b z < 0 . (6)

Then the set-valued mapping U ̄ : R n ⇉ R defined by U ̄ z ≜ u ∈ R : a z u + b z ≤ 0 is non-empty for all z ∈ R n and the controller u ∗ z ≜ arg min u ∈ R u − u n o m z 2 s . t . a z u + b z ≤ 0 (7) is locally Lipschitz on R n , and, for any point z^∗ such that a z ∗ = 0 , there exists a neighborhood N z ∗ such that u * z = u n o m z for all z ∈ N z * . Furthermore, the controller in Eq. 7 has a closed-form solution given by u ∗ z = − b z a z a z u n o m z + b z > 0 u n o m z o t h e r w i s e . (8)

Note that there is no division by zero in Eq. 8 since Eq. 6 implies that u * z = u n o m z when a z = 0 . Also note that the claim regarding the neighborhood about points for which a z * = 0 does not hold in general if the inequality in Eq. 6 is changed to b z ≤ 0 .

In this section, the electric motor control input is designed to ensure that the safe set S = { z ∈ R 2 : e L ≤ e ≤ e H } is UGAS for a given z_2d > 0. Our development is based on the design procedure described in Isaly et al. (2020) and the theoretical results for hybrid systems in Maghenem and Sanfelice (2021) and Goebel et al. (2012), where Maghenem and Sanfelice (2021) considers barrier functions specifically. The safe set S is encoded by the barrier function candidate B e : R 2 → R defined as B e z ≜ 1 2 M z 1 e 2 β e − 1 (9) where β e ≜ e L 2 e ≤ 0 e H 2 e > 0 .

Equivalent to the original definition, we have S = { z ∈ R 2 : B e z ≤ 0 } . The barrier function is designed to be continuously differentiable while encoding the potentially asymmetric (about e = 0) boundary of the set S .

While barrier functions are typically associated with forward invariance, they are naturally extensible to enforcing the stronger property of asymptotic stability. Asymptotic stability is beneficial for real-world applications since it guarantees robustness to perturbations from the safe set. For continuous-time systems, the stronger asymptotic stability condition is embodied in the definition of a zeroing CBF (ZCBF) Xu et al. (2015). Inspired by the ZCBF approach in our preliminary work in Isaly et al. (2020) and the work regarding CLFs for hybrid systems in Sanfelice (2013), we constrain the control input according to the so-called regulation map U ̃ : R 2 ⇉ U defined as U ̃ z ≜ u ∈ U : ∇ B e z , f ≤ − γ e e , ∀ f ∈ F u z , u , (10) where γ e e ≜ k b 1 e 2 β e − 1 , (11) and k_b1 > 0 is a control gain. In the stability analysis of Section 4, we show that selecting a continuous controller from U ̃ ensures that B_e acts as a Lyapunov function for the closed-loop dynamics outside the set S . In particular, the condition in Eq. 10 will be used to apply Proposition 3.27 of Goebel et al. (2012) to conclude that S , redefined to include some additional states, is UGAS.

Remark 2. The definition of a ZCBF requires that γ e z ≜ α B e z for an extended class K function a Xu et al. (2015). However, requiring dependence of γ_e on the barrier function candidate can be restrictive, and is not necessary to obtain theoretical guarantees. Because the value of the function M in B_e is unknown and cannot be implemented in a control law, we use γ_e instead of the typical ZCBF-based selection when defining U ̃ in Eq. 10. For both ZCBFs and the choice of γ_e in Eq. 11, the mapping in Eq. 10 enforces asymptotically stabilizing conditions on the barrier function at states outside the safe set.

The regulation map in Eq. 10 is not directly useful for design purposes because uncertainty in the dynamics prevents computation of the inequality (i.e., constraint) used to define U ̃ . In the rest of this section, we develop a new regulation map based on a computable constraint that is implementable in a QP like Eq. 7. The resulting QP-based controller is a locally Lipschitz selection of U ̃ . To compensate for the uncertainty introduced in Eq. 10 by F_u, we employ Lyapunov-based robust control techniques to develop a worst-case upper bound of the inner product in Eq. 10. For any z , u ∈ C u and f ∈ F u z , u , ∇ B e z , f ∈ 1 2 ∇ M z 1 z 2 e 2 β e − 1 + 1 β e e τ u z , u − τ F z . (12)

The product 1 β e e τ F z contains the term 1 β e e V p z z 2 = 1 β e e V p z e + V p z z 2 d , leading to a cancellation with the term 1 2 ∇ M z 1 z 2 e 2 β e since 1 2 ∇ M z 1 z 2 = V p z . Using Properties 2-7, it can then be shown that the unknown terms in Eq. 12 are upper bounded, for some constants c₁ − c₃, as ∇ B e z , f ≤ C e e + 1 β e e τ u z , u , (13) for each z , u ∈ C u and each f ∈ F u z , u , where C e e ≜ c 1 + c 2 e + c 3 e 2 .

In Eq. 13, the function τ_u depends on the motor control input u_e and the muscle control inputs u_M. Because the value of the subsequently designed muscle control input will jump at discrete instances, it is desirable to decouple the motor input from the muscle input to ensure the continuity of the motor controller. A continuous motor controller will be more predictable and comfortable for the rider. Using Property 8, there exists a constant c_M > 0 such that 1 β e e τ u z , u = 1 β e e c e u e + τ F E S z , u M ≤ c e β e e u e + c M e , (14) for all z ∈ R 2 , u e ∈ R , and u M ∈ U M . Using the definition of U M in Section 2, Eq. 14 applies to any muscle controller for which the inputs are bounded by the positive constant u ̄ . One can then define K e e ≜ k 1 + k 2 e + k 3 e 2 . (15)

Selecting ⁵ k 1 ≥ c 1 , k 2 ≥ c 2 + c M , k 3 ≥ c 3 , (16) implies that C e e + 1 β e e τ u z , u ≤ K e e + c e β e e u e (17) for all z ∈ R 2 , u e ∈ R , and u M ∈ U M . Since Eq. 17 is an upper bound for Eq. 13, we define a new regulation map U ̄ e : R ⇉ R as U ̄ e e ≜ u e ∈ R : K e e + c e β e e u e ≤ − γ e e . (18)

The following result summarizes the preceding development and explains the utility of U ̄ e .

Proposition 1. Assume k₁, k₂, and k₃ satisfy the gain conditions in Eq. 16 and z_2d > 0. Then, for any z ∈ R 2 and u M ∈ U M , if u e ∈ U ̄ e e , it follows that u e , u M ∈ U ̃ z .

The constraint used to define U ̄ e in Eq. 18 can be written in the generic form of Lemma 1. Additionally, since the terms in Eq. 18 are no longer uncertain, the constraint can be enforced with the following implementable QP: u e ∗ e ≜ arg min u e ∈ R u e − u e n o m e 2 s . t . K e e + c e β e e u e ≤ − γ e e , (19) where u e n o m : R → R is any locally Lipschitz nominal controller. According to Lemma 1, with b z = K e e + γ e e , the controller is feasible if e = 0 implies that K e 0 + γ e 0 = k 1 − k b 1 < 0 . (20)

Since the parameters in Eq. 20 are user-selected, they can be designed to ensure the inequality holds. Given this gain condition, the controller has the properties described in Lemma 1, in particular, it is continuous. The closed-form solution to Eq. 19 can be developed from Eq. 8. The controller is implementable in either form but the closed-form solution is computationally faster and does not require an optimization package. Note that the piecewise linear function e ↦ c e / β e e is locally Lipschitz, from which we derive local Lipschitz continuity of the controller.

Remark 3. During the experiments in Section 5, we investigated constant nominal controllers u e n o m ≜ u 0 for some u 0 ∈ R . When u₀ = 0, the controller is a minimum norm controller and the motor is off whenever possible while still ensuring safety. When the rider requires additional assistance, selecting a positive u₀ leads to the motor being biased to assist, while a negative u₀ leads to additional resistance and a more challenging training program. However, there is no theoretical obstacle to using a more complex nominal controller (e.g., one that tracks power).

To describe the FES control input, we define a concatenated state vector x ≜ z , σ M , τ ∈ X , where σ M ∈ 0,1 6 is a vector of switching signals σ m ∈ 0,1 defined for each muscle m ∈ M , τ ∈ R is a timer variable, and X ≜ R 2 × 0,1 6 × R is the state space. The stimulation input to the rider’s muscle groups u M ∗ : X → R 6 is defined for each muscle as ⁶ u m ∗ x ≜ sat u ̄ + σ m u F E S ∗ e , (21) where u F E S ∗ : R → R will be defined subsequently. The switching signals are updated at jumps according to the rule σ m + = 1 z 1 ∈ Q m 0 z 1 ∉ Q m , (22) where Q m was defined in Section 2. The update rule specifies that the rider’s muscles are stimulated in regions where they are able to produce positive torque. The rule for σ_m and the timer variable τ will be used to define a hybrid system in Section 4.

As discussed in Section 3.1, it is generally not possible to maintain the cadence in the set S F E S = { z ∈ R 2 : e ≥ e F E S } because the stimulation input u m ∗ is limited in magnitude and only intermittently available. The function u F E S ∗ in Eq. 21 represents a selection of the input that would render the set S F E S asymptotically stable in the absence of these obstacles (and subject to some gain conditions). The development of u F E S ∗ is very similar to the one in Section 3.2 and is therefore omitted to avoid redundancy. Moreover, we do not make claims in the forthcoming stability analysis regarding S F E S . The FES input is defined by the following QP: u F E S ∗ e ≜ arg min u F E S ∈ R u F E S − u F E S n o m e s . t . K F E S e + 1 β 2 e e u F E S ≤ − γ F E S e . (23) where u F E S n o m : R → R is any locally Lipschitz nominal controller, K F E S e ≜ k 4 + k 5 e + k 6 e 2 , γ F E S e ≜ k b 2 e 2 β 2 e − 1 , and β 2 e ≜ e F E S 2 e ≤ 0 e H 2 e > 0 . When e = 0, the feasibility condition (6) in Lemma 1 requires that K F E S 0 + γ F E S 0 = k 4 − k b 2 < 0 , (24) under which the function u F E S ∗ has the properties described in the lemma. Similar to the previous section, the parameters in Eq. 24 are user-selected and can be designed to ensure the inequality holds.

When the nominal controller in Eq. 23 is set to u F E S n o m ≜ 0 , then from the closed-form solution in Eq. 8 it can be determined that u F E S ∗ e ≤ 0 for all e ≥ 0. In this case, because of the use of the saturation function in Eq. 21, it follows that u m * x = 0 for all x ∈ X such that e ≥ 0. The inclusion of a nominal controller in the QP defining u F E S * gives the operator flexibility to provide stimulation at points where e ≥ 0. When using a nominal controller with u F E S n o m e > 0 for e ≥ 0, FES stimulation produces torque to increase the cadence above the setpoint z_2d. However, it is always the case that u F E S * e → 0 as e → e_H. By combining nominal assistance from FES with nominal resistance from the electric motor, a more intense training program can be designed where the rider must work against resistive torque from the electric motor to stay near the setpoint. We provide experimental results for a higher intensity configuration of the control system in Section 5.

The experimental testbed consisted of a stationary recumbent tricycle (TerraTrike Rover) with a 250 W, 24 V motor (Unite Motor Co.) coupled to the drive chain as described in Bellman et al. (2017), Section 5.1. To measure position and cadence, an optical encoder with an angular resolution of 20,000 pulses per revolution (US Digital H1) was mounted to the crank using spur gears. The motor was actuated using an Advanced Motion Controls ¹⁰ motor driver and current-controlled power supply. Stimulation was delivered to the rider’s quadricep, hamstring, and gluteal muscle groups via self-adhesive electrodes provided compliments of Axelgaard Manufacturing Co., Ltd. A current-controlled stimulator (Hasomed Rehastim) delivered symmetric, rectangular, and bi-phasic pulses at fixed amplitude (90 mA, 80 mA, and 70 mA for the quadriceps, hamstrings, and gluteals, respectively) and frequency (60 Hz), while the pulse width was used as the control input. A desktop computer running real-time control software (QUARC integrated with Simulink) was used to interface the controllers and hardware through a data acquisition board (Quanser QPIDe) with a sampling rate of 1,000 Hz. For additional safety, an emergency stop switch was mounted on the cycle to allow the participant to end the experiment if required.

The primary testing procedure (Protocol A) consisted of 180-s tests for each of the three configurations (barrier function, 3M, and volition-only) under consideration. The volitional pedaling trial was always first, followed by a random selection of either the barrier function or 3M controller. The riders were asked to track a setpoint of z_2d = 50 RPM. The safe set boundary for the barrier function controller was encoded by e_L = − 5 RPM and e_H = 5 RPM, while e_FES = − 3 RPM. The inactive region for the 3M controller was 48–52 RPM, which is comparable to the range 50–55 RPM that was used for the experiments in Rouse et al. (2020). The rider was shown a live plot of their cadence featuring a visible indication of the setpoint. Due to differences between the barrier function and 3M controllers, participants were not shown the boundaries of the safe set ¹¹ .

The 180 s tests started with a 20 s ramp-up phase where the rider sat passively while the motor brought their cadence to the setpoint. To ensure that the presented data represented steady-state operation, the ramp-up and an additional 20 s after it was excluded from each dataset in post-processing. For each configuration, there was a separate warm-up run before the recorded session so the rider could become accustomed to the controller.

Measurements of the position of the rider’s legs with respect to the cycle were used to determine the regions of effective torque transfer Q m for each muscle (see Bellman et al. (2017), Section 5.2) for more details). The cycle was initially operated at 50 RPM and open-loop stimulation was applied to one muscle group at a time to determine the comfort limit u ̄ for each muscle. The FES inputs are scaled by the comfort threshold in addition to being saturated ¹² . The nominal controllers were u e n o m = u F E S n o m = 0 . The control gains were adjusted by plotting the control inputs as a function of the cadence error, which produced a visualization of the regions of applied control effort. Small adjustments to the gains were made for each participant based on their preferences during the warm-up run, which is a cause for variation in the data between participants. Some detailed discussion about the effect of the barrier function controller gains on performance is provided in Isaly et al. (2020) (see Remark 1 and Section 5.4).

Additional trials were conducted to highlight unique aspects of the barrier function controller, which were performed with only one participant. Protocol B was designed to prioritize power output from the rider over cadence tracking. For this alternative trial, the width of the safe range was larger, with e_L = − 12 RPM, e_H = 10 RPM, and e_FES = − 6 RPM. Because the rider (Participant #2) was able-bodied, we chose to make the program more challenging by adding nominal controllers with u e n o m < 0 and u F E S n o m > 0 , which means that near the setpoint the electric motor produced resistive torque while FES provided assistance. The boundaries e_L and e_H were displayed on-screen for this trial. In Protocol C, Participant #1 was asked to provide no volitional effort for both the 3M and barrier function controllers (Figure 4). Because of problems with the 3M controller for this trial, we did not proceed with testing the no-volition configuration on other participants.

Table 1 shows relevant statistics for the three configurations tested for Protocol A, including the average and standard deviation of the cadence, percentage of the trial duration for which FES was actively stimulating, and time spent outside of the safe set S . Two integrals of the electric motor input are given to distinguish resistive torque, for which the rider must pedal harder to compensate, and assistive torque, which implies work done by the motor and not the rider. The barrier function controller produced the lowest standard deviation in cadence for each participant and led to greater FES usage, but generally used more assistive torque from the motor than the 3M controller. The 3M controller produced less assistive torque because the electric motor was off in regions of the crank cycle where FES was active. The discrete switching between motor and FES, along with switching on the boundary of the inactive zone, caused numerous discontinuities in the 3M motor controller; Table 1 shows an average of 493 switches per trial. For all participants, the riders’ cadence took values in the range 40.7–56.1 RPM for the 3M controller, 45.4–55.3 RPM for the barrier function controller, and 40.5–59.2 RPM for uncontrolled pedaling. The barrier function controller constrained each rider’s cadence to the user-defined range of 45–55 RPM for all but a negligible amount of time; an average of six sampled data points or approximately 0.004% of the trial duration. Segments of the trials for three randomly selected participants are shown in Figure 2. A zoomed view featuring the FES stimulation input for a single participant is shown in Figure 3.

Figure 4 shows the trials using Protocol C, where Participant #1 provided no volitional effort. The barrier function controller was still able to keep the rider’s cadence within the safe set, while the 3M controller caused large oscillations; the mean ± SD cadence for the 3M controller was 43.74 ± 4.94 RPM. The cadence dropped as low as 31.54 RPM during the 3M trial because the electric motor was inactive in the shaded regions of Figure 4. When the motor was next switched on, it compensated using control action with a maximum magnitude of 17.74 A, which was relatively large compared to a maximum of 8.13 A for the 3M controller during the Protocol A trials.

The results for Protocol B, where there were nonzero nominal controllers and a wider safe range, are displayed in Figure 5 and Table 2. There was high utilization of the electric motor to produce resistive torque, but low assistive torque production. The assistive torque was smaller than the Protocol A average and, in particular, was smaller than Participant #2’s results. The resistive torque was large and FES was more active for Protocol B due to the design of the nominal controllers. The electric motor deviated from its nominal value for only 7.7% of the trial duration and was providing assistance for 4.1% of the trial. The cadence standard deviation was higher than uncontrolled volitional pedaling, which was most likely because the control inputs were actively pushing the rider away from the setpoint. The assistive torque from the barrier function controller in Protocol B was comparable to the 3M controller in the Protocol A trials.

The results for Protocol A demonstrate that the barrier function controller can assist a rider in tracking the cadence setpoint while constraining their cadence within a user-defined range. Figure 3 shows that the control inputs ramp up before the cadence reaches the boundaries defined by e_H, e_L, and e_FES, yet are inactive when the cadence is near the setpoint. Such a ramp-up is demonstrative of how barrier function methods can ensure a gradual transition from a nominal controller, which should be active on the interior of the safe set, to an invariance-ensuring controller on the boundary of the safe set. For Protocol A, the width of the nominal control range was fairly small, while the width was increased for Protocol B.

In some situations, increasing the training intensity for the rider will be preferred over improved cadence tracking. The relevant statistics are the assistive and resistive torques produced by the electric motor. Higher assistive torque indicates that less power is being produced by the rider. The barrier function controller is versatile and can be tuned to provide more or less interference from the control inputs. Figure 5 shows a trial using Protocol B where power output from the rider was prioritized by allowing larger cadence errors. The system’s operator can allow larger cadence errors by selecting a wider safe range, which also facilitates the design of a wider nominal control range. The use of nominal controllers for Protocol B intensified the training by providing more FES stimulation and resistance from the motor. The control inputs were at their nominal values for a significant portion of the experiment, validating that a wider safe range leads to less modification of the nominal inputs. The nominal inputs can alternatively be designed so there is less resistance from the motor or less control input overall. The fact that the assistive torque from the barrier function controller was comparable to the 3M controller suggests that staggering FES assistance before motor assistance, as in the barrier function controller, is a viable alternative to discrete switching between FES and motor control, as in the 3M controller.

A continuous motor controller is more comfortable for the rider. Continuity is the primary difference between the barrier function and 3M controller. The 3M controller is discontinuous whenever the cadence crosses the boundary of the inactive zone, or the angular position crosses the boundary of the FES stimulation regions Q m . The resulting large number of switches for the 3M motor controller is quantified in Table 1. A particular advantage of the motor being active in the Q m regions is that the barrier function controller is effective even when the rider produces little or no volitional effort, as shown in the trials using Protocol C (Figure 4). The 3M controller caused large oscillations in the rider’s cadence during Protocol C due to the discrete switching between FES and motor control. Since FES cannot always produce enough torque on its own, the cadence dropped in the FES stimulation regions, which was then met with large control effort from the motor upon exiting the region.

There were some small deviations from the safe set because the asymptotic stability of S is contingent on the selection of control gains, which must be large enough to compensate for the rider’s volitional effort and other dynamic effects. In practice, designing the gains to account for large volitional contributions from the rider leads to overly constraining the rider during normal operation. Designing controllers for asymptotic stability, rather than the weaker property of forward invariance, helps mitigate these effects. A property known as input-to-state stability, which is a consequence of the UGAS result in Theorem 1, can be interpreted as guaranteeing that some nearby set is asymptotically stable despite unaccounted-for disturbances of bounded magnitude Xu et al. (2015); Cai and Teel (2009). Thus, the control gains can be relaxed to favor comfortable and effective therapy while still ensuring that the cadence remains nearby the safe set.