Robust Control of the Human Trunk Posture Using Functional Neuromuscular Stimulation: A Simulation Study

Abstract

The trunk movements of an individual paralyzed by spinal cord injury (SCI) can be restored by functional neuromuscular stimulation (FNS), which applies low-level current to the motor nerves to activate the paralyzed muscles to generate useful torques, to actuate the trunk. FNS can be modulated to vary the biotorques to drive the trunk to follow a user-defined reference motion and maintain it at a desired postural set-point. However, a stabilizing modulation policy (i.e., control law) is difficult to derive as the biomechanics of the spine and pelvis are complex and the neuromuscular dynamics are highly nonlinear, nonautonomous, and input redundant. Therefore, a control method that can stabilize it with FNS without knowing the accurate skeletal and neuromuscular dynamics is desired. To achieve this goal, we propose a control framework consisting of a robust control module that generates stabilizing torques while an artificial neural network-based mapping mechanism with an anatomy-based updating law ensures that the muscle-generated torques converge to the stabilizing values. For the robust control module, two sliding-mode robust controllers (i.e., a high compensation controller and an adaptive controller), were investigated. System stability of the proposed control method was rigorously analyzed based on the assumption that the skeletal dynamics can be approximated by Euler–Lagrange equations with bounded disturbances, which enables the generalization of the control framework. We present experiments in a simulation environment where an anatomically realistic three-dimensional musculoskeletal model of the human trunk moved in the anterior– posterior and medial–lateral directions while perturbations were applied. The satisfactory simulation results suggest the potential of this control technique for trunk tracking tasks in a typical clinical environment.

1 Introduction

Spinal cord injury (SCI) is a serious condition that can disrupt the communication between the brain and sensorimotor nerves, which results in the loss of body functionality [1]. Up to 2020, around 294,000 people in the United States are reported to have SCI, and the number is increasing by approximately 17,810 per year [2]. Therefore, restoring the body functionality for individuals with SCI is of practical significance.

In recent years, a variety of exoskeleton devices, which have the potential of assisting in the daily life of individuals with SCI, have been developed. They can enhance human power, i.e., performance augmenting exoskeleton, or directly provide external powers, i.e., active orthoses, to achieve functional tasks [3,4], such as standing up, stair climbing, walking (e.g., ReWalk [5], EXPOS [6], Ekso [7], HAL [8]), arm-reaching (e.g., Exorn [9]), grasping (e.g., SaeboFlex [10]), or full-body supporting (e.g., FORTIS/HULC [3]). Trunk functionality, which plays a critical role in daily life [11] as it facilitates the use of all the extremities and the head, can also be assisted with performance augmenting exoskeleton devices, e.g., SPEXOR [12]. However, active orthoses devices, which can facilitate trunk movement have not been developed. This is perhaps due to the high complexity, flexibility, and uncertainty of the human trunk system, which makes the design difficult. Therefore, a new approach is needed that can actively control human trunk to achieve functional movements.

Functional neuromuscular stimulation (FNS) is a technique that activates paralyzed muscles by applying low-level current to the motor nerves to generate muscle forces, which can be coordinated to achieve many functional movements (e.g., grasping and reaching [13,14], standing [15,16], walking [17–19], cycling [20]). It is proven to be an effective way of transmitting forces into the neuromusculoskeletal dynamics after SCI.

Applications of FNS to human trunk systems are also being extensively investigated [21–29]. The rehabilitation function of FNS has been investigated in Ref. [23], where clinical results show that FNS can enhance rehabilitation training and increase trunk reaching limits. The work in Ref. [22] shows that FNS can enhance trunk stability with a simple proportional–derivative controller. This conclusion was further verified with five human subjects in Ref. [21]. These FNS-based control applications have mainly been focused on maintaining trunk balance in the erect posture. Recently, a user-controlled FNS technique has been developed [30] for reaching task. However, trunk autotracking control (i.e., smoothly driving the trunk to follow a predefined trajectory), as shown in Fig. 1, has not yet been achieved.

Fig. 1.

The figure shows how the human trunk motions are generated by FNS-induced torques. Principally, activating different muscle nerves can result in motions at different directions, which also obey the rules of vector composition.

The lack of robust control methods to coordinate the actions of the stimulation-activated paralyzed muscles is likely the reason that implementing systems to generate desired trunk movement trajectories have been difficult to accomplish. This is due, in part, to that the skeletal system has complex rigid body dynamics in which several bones (T1-T12, L1-L5, S1-S5) are coupled together via three degree-of-freedom joints [29]. This also causes the reliable estimation of many parameters (e.g., total weight, center of mass, moment of inertia) to be very difficult. Moreover, it is not possible to directly apply the desired torques to the joints since they must be generated via stimulating the multiple nerves innervating corresponding muscles at each vertebral segment [31–34] (e.g., erector spinae, rectus abdominis, quadratus lumborum). As the neuromuscular dynamics of these muscles are not known exactly, the corresponding sequence of stimulation that can generate the desired torques is also not easily ascertained. Furthermore, muscles are highly nonlinear, nonautonomous, and input-redundant, complicates the identification of their exact characteristics. Therefore, developing a control method that can stabilize the trunk tracking tasks via FNS without a full knowledge of the skeletal and neuromuscular dynamics will be a useful contribution to rehabilitation of individuals with SCI, and enhance their independence and ability to control objects in the environment.

In this paper, we propose a control framework where a robust control module provides stabilizing torques while an adjustable mapping based on an artificial neural network (ANN) structure ensures the muscle generated biotorques converge to the desired stabilizing values. The robust control module is derived based on the assumption that the skeleton is a two-link inverted pendulum model as the hips and one lumbar joint are considered. The feedback terms in the robust control module ensure that the bounded unmodeled errors and other perturbations are rejected. The adjustable mapping consists of a self-organizing neural network (SONN) [35,36] and a two-layer correction neural network (CoNN). The SONN is implemented to provide a basic mapping in a lookup-table manner, which can also solve the input-redundancy. The CoNN, which is adjusted using an anatomy-based updating law, supplies correction signals to drive FNS-induced torques to follow the controller-computed stabilizing torques. For the robust control module, two sliding-mode robust controllers, i.e., a high compensation controller and an adaptive controller, were investigated. Rigorous stability analysis of the integrated control loop is demonstrated.

To test the potential of the controller to stabilize an actual human trunk, a simulation was run on a more realistic three-dimensional musculoskeletal model of the trunk created using the opensim musculoskeletal modeling software [37]. In the simulation, the hip and lumbar joints were actuated to move the trunk to new postures while bounded disturbances were added, which simulated object acquisition and manipulation tasks. The satisfactory performance verified the effectiveness of the proposed framework.

The contribution of this paper can be summarized as:

• A robust control framework was developed to control the human trunk system to perform tracking tasks without knowing the neuromusculoskeletal dynamics.

• An ANN mapping, which consists of a SONN and a CoNN, is derived for transmitting the controller-computed torques to the skeletal (rigid-body) dynamics of the trunk.

• Specifically designed controllers with feedforward terms (based on the approximation of the trunk skeletal model) and feedback terms (based on sliding-mode terms) are proven and shown able to stabilize the control process in the presence of disturbances. Results obtained in a realistic simulation environment validated the power of the control method.

As regulating or tracking trunk posture of a human subject is considered to be complicated, expensive, and safety-sensitive, it is necessary to analyze the anatomical system, develop control methods, and validate them in simulation before the actual experimental implementation. This paper provides the preliminary steps, which can enable human-in-the-loop experiments in the future.

2 Human Trunk System

2.1 Anatomical Consideration.

The human trunk system can be modeled as a skeletal subsystem plus a neuromuscular subsystem, which are shown in Fig. 2. In this paper, we mainly consider the rotations of the pelvis about the hip joints and the trunk about the S1–L5 joint. The trunk link lumps the whole spine, thorax, arms, head, and neck into a single body. Therefore, system rigid-body dynamics can be approximately modeled as a two-link inverted pendulum. The hips are modeled as a one-degree-of-freedom joint (defining pelvic pitch) while the S1–L5 joint is modeled as a two-degree-of-freedom joint (defining trunk pitch and trunk roll). Thus, the skeletal system is characterized by a three-degree-of-freedom system. The neuromuscular model presents the dynamics that map the FNS to the muscle-generated torques about the two joints. The details are given in the following content.

Fig. 2.

This figure shows the human trunk model. (a) Skeletal model: it shows the main links and regions. (b) Neuromuscular model: it shows how the muscle activation causes the torques that generate joint motions.

2.2 Skeletal Dynamics.

The generalized Euler–Lagrange equations that model the n-degree-of-freedom skeletal subsystem can be expressed as

$$M(\mathit{q})\ddot{\mathit{q}}+C_m(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+G(\mathit{q})+F(\mathit{q},\dot{\mathit{q}})+\mathit{d}=\mathit{\tau }$$ (1)

where the terms $\mathit{q},\dot{\mathit{q}},\ddot{\mathit{q}}\in {ℝ}^n$ are the joint angular position, angular velocity, and angular acceleration vectors, respectively. The matrix $M(\mathit{q})\in {ℝ}^{n\times n}$ denotes the generalized inertia matrix, $C_m(\mathit{q},\dot{\mathit{q}})\in {ℝ}^{n\times n}$ denotes the Centripetal–Coriolis matrix, $G(\mathit{q})\in {ℝ}^n$ denotes the gravity vector, $F(\mathit{q},\dot{\mathit{q}})\in {ℝ}^n$ denotes the passive viscoelastic vector, $\mathit{d}\in {ℝ}^n$ represents any disturbances on the system. The term $\mathit{\tau }\in {ℝ}^n$ is the torque vector. Some typical properties of a rigid-body dynamics are given Ref. [38].

Property 1: The inertia matrix is symmetric, positive definite, bounded, and invertible, i.e., $m_1||\xi |{|}^2\le {\xi }^{T}M\xi \le m_2||\xi |{|}^2$.

Property 2: The dynamics in Eq. (1) follows the skew symmetric property, i.e., $\dot{M}-2C_m=0$.

Assumption 1: The disturbance term $\mathit{d}\in {ℝ}^n$ is bounded as $||\mathit{d}||\le \overline{d}$, where $\overline{d}\in ℝ$ denotes the upper limit.

In this paper, we consider the pelvic pitch, q₁, lumbar pitch, q₂, and lumbar roll, q₃ (see Fig. 3), as the main generalized coordinates. Therefore, there are three degree-of-freedom, i.e., n = 3, and the vector $\mathit{q}={[\begin{array}{cc}{q}_1& q_2\end{array}{q}_3]}^T$. The physical constraints limit the joint angles within a reasonable region, i.e., $\mathit{q}\in {\mathcal{Q}}_0\subseteq {ℝ}^3$, $(\mathit{q},\dot{\mathit{q}})\in {\mathcal{Q}}_1\subseteq {ℝ}^6$.

Fig. 3.

Joint angles in human trunk model

2.3 Neuromuscular Dynamics

2.3.1 Dynamics and Estimation Representation.

Our musculoskeletal model of the human trunk is actuated by 26 muscle groups that can affect trunk movements in the sagittal and coronal planes. The muscles are bilaterally, erector spinae (ES), rectus abdominis (RA), external obliques (EO), internal obliques (IO), quadratus lumborum (QL), three branches of gluteus maximus (GM), adductor magnus (AM), semimembranosus (SM), sartorious (SA), iliacus (IL), and psoas (PS). In the subsequent discussion, the name of each muscle is prepended with the letters L for left and R for right. With this, the vector of muscle activations is defined as:

$$\begin{array}{cccccc}\mathit{a}=& [a_{\mathit{RES}}& a_{\mathit{LES}}& a_{\mathit{RRA}}& a_{\mathit{LRA}}& a_{\mathit{REO}}\\ & a_{\mathit{LEO}}& a_{\mathit{RIO}}& a_{\mathit{LIO}}& a_{\mathit{RQL}}& a_{\mathit{LQL}}\\ & a_{RG{M}_1}& a_{LG{M}_1}& a_{RG{M}_2}& a_{LG{M}_2}& a_{RG{M}_3}\\ & a_{LG{M}_3}& a_{\mathit{RAM}}& a_{\mathit{LAM}}& a_{\mathit{RSM}}& a_{\mathit{LSM}}\\ & a_{\mathit{RSA}}& a_{\mathit{LSA}}& a_{\mathit{RIL}}& a_{\mathit{LIL}}& a_{\mathit{RPS}}\\ & a_{\mathit{LPS}}]& & & & \end{array}$$

Each $a\in ℝ$ in $\mathit{a}\in {ℝ}^{26}$ is normalized as $a\in [0,1]$, where 0 is the current that cause the minimum torque and 1 the maximum torque.

We define the neuromuscular dynamics to be $\mathcal{T}(·):\hspace{0.17em}{ℝ}^{26}\mapsto {ℝ}^3$ that transfers the muscle activations to joint torques, and assume that there is a mapping $\mathcal{M}(·):\hspace{0.17em}{ℝ}^3\mapsto {ℝ}^{26}$ such that

$$\mathit{\tau }=\mathcal{T}(\mathit{a})=\mathcal{T}(\mathcal{M}(\mathit{\tau }))$$ (2)

The mapping $\mathcal{M}(·)$ in Eq. (2) presents an ideal inverse dynamics of $\mathcal{T}(·)$. It may not be unique and is unknown. In practice, it should be estimated by a feed-forward activation mapping as $\widehat{\mathcal{M}}(·)$. Assuming there is an ideal correction mapping $f_c(·):\hspace{0.17em}{ℝ}^3\mapsto {ℝ}^{26}$ makes

$$\mathcal{M}(\mathit{\tau })=\widehat{\mathcal{M}}(\mathit{\tau })+f_c(\mathcal{T}(\mathcal{M}(\mathit{\tau }))-\mathcal{T}(\widehat{\mathcal{M}}(\mathit{\tau })))$$ (3)

The function $f_c(·)$ in Eq. (3) can be decomposed as

$$f_c=f_N+\mathit{\epsilon }$$

where $\mathbf{\epsilon }\in {ℝ}^{26}$ is an unknown residual error vector, $\widehat{\mathcal{M}}$ can be designed using a static feedforward neural network (NN), and f_N can be modeled by a two-layer NN [38]

$$f_N={\sigma }_w(W^T{\sigma }_v(V^T\mathit{x}))$$

where $W\in {ℝ}^{h\times 26}$ and $V\in {ℝ}^{l\times h}$ are the ideal NN weight matrices, ${\sigma }_w\in {ℝ}^{26}$ and ${\sigma }_v(·)\in {ℝ}^h$ are activation functions, and $\mathit{x}={[\mathit{1},z]}^T\in {ℝ}^l$, where z belongs to a compact set, is the input to the NN, which can be selected as long as it is formed based on $\mathcal{T}(\mathcal{M}(\mathit{\tau }))-\mathcal{T}(\widehat{\mathcal{M}}(\mathit{\tau }))$ (details can be seen in Ref. [39])

The estimation of f_N is defined as ${\widehat{f}}_N$, which can be written as [38]

$${\widehat{f}}_N={\sigma }_w({\widehat{W}}^T{\sigma }_v({\widehat{V}}^T\mathit{x}))$$

where $\widehat{W}$ and $\widehat{V}$ are the estimated NN weights. Therefore, we write

$${\mathit{\tau }}_m=\mathcal{T}(\widehat{\mathcal{M}}(\mathit{\tau })+{\sigma }_w({\widehat{W}}^T{\sigma }_v({\widehat{V}}^T\mathit{x})))$$ (4)

where ${\mathit{\tau }}_m$ denotes the actual torque generated by FNS. The physical meaning of Eq. (4) is that even though the desired torque is $\mathit{\tau }$ the actual torque generated by FNS is ${\mathit{\tau }}_m$. We define the error term ${\mathit{\tau }}_d=\mathit{\tau }-{\mathit{\tau }}_m$, which equals 0 when $\mathcal{M}(\mathit{\tau })=\widehat{\mathcal{M}}(\mathit{\tau })$. We also present the following assumptions

Assumption 2: The error in the feed-forward activation mapping estimation is bounded, i.e., $\forall \mathit{\tau }\in {ℝ}^3$ we have $||\mathcal{M}(\mathit{\tau })-\widehat{\mathcal{M}}(\mathit{\tau })||\in {\mathcal{L}}_{\infty }$.

Assumption 3: There exists an estimation, ${\widehat{f}}_N$, of the ideal NN, f_N, that forms ${\mathit{\tau }}_m=\mathcal{T}(\widehat{\mathcal{M}}(\mathit{\tau })+{\widehat{f}}_N)$, such that for all x with initial NN weights ${\widehat{W}}^T(0)$ and ${\widehat{V}}^T(0)$ we have $||{\mathit{\tau }}_d||\le {\overline{\tau }}_d$. Here ${\overline{\tau }}_d\in ℝ$ denotes the upper bound of $||{\tau }_d||$.

2.3.2 Anatomy-Based Updating Algorithm.

The NN ${\widehat{f}}_N$ is adjustable so that we can derive an updating law for its weights to ensure that ${\mathit{\tau }}_m$ converges to $\mathit{\tau }$ locally. However, the updating law contains the neuromuscular dynamics if gradient-based search (under backpropagation structure) is applied. Since the neuromuscular dynamics are not fully known, the application of the updating law becomes invalid. This section shows that an anatomy-based direction matrix can restore the viability of the updating law even with the uncertainty in the neuromuscular dynamics.

Defining the error objective as

$$E=\frac{1}{2}{\mathit{\tau }}_d^T{\mathit{\tau }}_d$$ (5)

and combining with Eq. (4), we obtain

$$\begin{array}{c}\frac{\partial E}{\partial \widehat{W}}={\left({\mathit{\tau }}_d^T{\mathcal{T}}^{\prime }{\sigma }_w^{\prime }\right)}^T{\sigma }_v^T\\ \frac{\partial E}{\partial \widehat{V}}={\left({\mathit{\tau }}_d^T{\mathcal{T}}^{\prime }{\sigma }_w^{\prime }\widehat{W}{\sigma }_v^{\prime }\right)}^T{\mathit{x}}^T\end{array}$$

which means that if an NN updating law is chosen as the following the error E is decreasing with time:

$$\begin{array}{c}\dot{\widehat{W}}=-F{\sigma }_v{\mathit{\tau }}_d^T{\mathcal{T}}^{\prime }{\sigma }_w^{\prime }\\ \dot{\widehat{V}}=-G\mathit{x}({\mathit{\tau }}_d^T{\mathcal{T}}^{\prime }{\sigma }_w^{\prime }{\widehat{W}}^T{\sigma }_v^{\prime })\end{array}$$ (6)

where F and G are symmetric matrices and all entries in them are positive. Therefore, the updating law (6) can ensure that for ${\widehat{W}}^T(t)$ and ${\widehat{V}}^T(t)$ (t > 0)

$$||{\mathit{\tau }}_d(t)||\le {\overline{\tau }}_d$$ (7)

The mapping, $\mathcal{T}$, in Eq. (6), is actually the neuromuscular dynamics, which is highly complex and not completely known. Therefore, taking the derivative (i.e., computing ${\mathcal{T}}^{\prime }\in {ℝ}^{3\times 26}$) in Eq. (6), may be not plausible.

However, the direction of the updating (i.e., direction of $\dot{\widehat{W}}$ and $\dot{\widehat{V}}$) determines if the error decreases while the step size (i.e., magnitude of $\dot{\widehat{W}}$ and $\dot{\widehat{V}}$) only determines how far per step to move along the direction. In other words, the step size usually does not affect the convergence if it is sufficiently small but the direction of the updating has to be correct. To utilize this property, let's first write ${\mathcal{T}}^{\prime }$ as

$${\mathcal{T}}^{\prime }=|{\mathcal{T}}^{\prime }|°\mathcal{D}$$ (8)

where $°$ denotes Hadamard product, $|{\mathcal{T}}^{\prime }|\in {ℝ}^{3\times 26}$ is a matrix that consists of the magnitude of each entry, and matrix $\mathcal{D}\in {ℝ}^{26\times 3}$ denotes the direction of ${\mathcal{T}}^{\prime }$. The direction matrix $\mathcal{D}$ can be determined in a simple and heuristic way by examining the anatomical structure of the trunk musculoskeletal dynamics. For instance, when Semimembranosus is activated only pelvic pitch angle is mainly influenced, while Erector Spinae can drive both lumbar pitch and roll angles, which can be seen in Fig. 4.

Fig. 4.

Activating semimembranosus can cause the pelvic pitch to increase along the positive direction (counterclockwise if right side is pointing out of the paper); activating both the right (R) and left (L) erector spinae simultaneously can cause the lumbar pitch to increase, while activating R erector spinae increases the lumbar roll and L decreases lumbar roll

Therefore, $\mathcal{D}$ can be expressed as

$$\mathcal{D}={\left(\begin{array}{ccc}0& -1& 1\\ 0& -1& -1\\ 0& 1& 0\\ 0& 1& 0\\ 0& 0& 1\\ 0& 0& -1\\ 0& 0& 1\\ 0& 0& -1\\ 0& 0& 1\\ 0& 0& -1\\ -1& 0& 0\\ -1& 0& 0\\ -1& 0& 0\\ -1& 0& 0\\ -1& 0& 0\\ -1& 0& 0\\ -1& 0& 0\\ -1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\\ 1& 1& 0\\ 1& 1& 0\end{array}\right)}^T$$

Each column of this matrix corresponds to a movement direction; forward (+1) and backward (−1) in the sagittal plane and to the right (+1) and left (−1) in the coronal plane while each row corresponds to a muscle actuator. An actuator that is active in two or more directions will have nonzero entries in two or more rows of $\mathcal{D}$.

Thus, the updating law (6) becomes

$$\begin{array}{c}\dot{\widehat{W}}=-F{\sigma }_v{\mathit{\tau }}_d^T|{\mathcal{T}}^{\prime }|°\mathcal{D}{\sigma }_w^{\prime }\\ \dot{\widehat{V}}=-G\mathit{x}{\mathit{\tau }}_d^T|{\mathcal{T}}^{\prime }|°\mathcal{D}{\sigma }_w^{\prime }{\widehat{W}}^T{\sigma }_v^{\prime }\end{array}$$ (9)

According to the anatomy analysis, activation of each muscle is independent from others and the torques are linearly mapped from the activation [13,33], i.e.,

$${\tau }_i=\sum _{j=1}^m{\phi }_j^i(q_i^{(1)},q_i^{(2)},\dots ,q_i^{(n)},l_i^0)a_j$$

where τ_i denotes the torque at the $i^{\text{th}}$ joint, a_j is the activation of the $j^{\text{th}}$ muscle, $q_i^{(n)}$ is the $i^{\text{th}}$ joint angle with $n^{\text{th}}$ order, $l_i^0$ is the original length of the $i^{\text{th}}$ muscle fiber, and ${\phi }_j^i$ is a function with constant sign and ${\phi }_j^i=0$ when the $j^{\text{th}}$ muscle does not affect the joint. It implies that $|{\mathcal{T}}^{\prime }|$ does not contain activation.

By introducing $S\in {ℝ}^{h\times 3},\hspace{0.17em}B\in {ℝ}^{26\times h}$, $U\in {ℝ}^{l\times 3},\hspace{0.17em}P\in {ℝ}^{26\times l}$, we can rewrite Eq. (9) to

$$\begin{array}{c}\dot{\widehat{W}}=-S|{\mathcal{T}}^{\prime }|{\sigma }_w^{\prime }BF{\sigma }_v{\mathit{\tau }}_d^T\mathcal{D}\\ \dot{\widehat{V}}=-U|{\mathcal{T}}^{\prime }|{\sigma }_w^{\prime }PG\mathit{x}{\mathit{\tau }}_d^T\mathcal{D}{\widehat{W}}^T{\sigma }_v^{\prime }\end{array}$$ (10)

and due to the analysis above, the terms $S|{\mathcal{T}}^{\prime }|B$ and $U|{\mathcal{T}}^{\prime }|P$ can be absorbed into F and G, respectively (i.e., $F_t=S|{\mathcal{T}}^{\prime }|BF$ and $G_t=U|{\mathcal{T}}^{\prime }|PG$). Therefore, the updating law becomes

$$\begin{array}{c}\dot{\widehat{W}}=-F_t{\sigma }_v{\mathit{\tau }}_d^T\mathcal{D}{\sigma }_w^{\prime }\\ \dot{\widehat{V}}=-G_t\mathit{x}{\mathit{\tau }}_d^T\mathcal{D}{\sigma }_w^{\prime }{\widehat{W}}^T{\sigma }_v^{\prime }\end{array}$$ (11)

In the neuromuscular dynamics, the muscle activation only directly affects the torque. Therefore, even if there are dynamical couplings in the movements of the joints via induced acceleration [40], eight still holds.

3 Control Method Development

As outlined above, the human trunk system dynamics can be described by the equation

$$M(\mathit{q})\ddot{\mathit{q}}+C_m(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+G(\mathit{q})+F(\mathit{q},\dot{\mathit{q}})+\mathit{d}=\mathcal{T}(\mathit{a})$$ (12)

where the LHS is the Euler–Lagrange type dynamics with uncertainty, i.e., skeletal part, and the RHS is an unknown actuator function, i.e., neuromuscular part.

A feedback law for modulating the activation, a, in Eq. (12) ( ${\mathit{\tau }}_c=\mathcal{M}({\mathit{\tau }}_c)$in this case) is expected to provide stabilizing torque, ${\mathit{\tau }}_c$, to actuate the dynamics to drive q to follow ${\mathit{q}}_d$. But as $\mathcal{T}(·)$ is not completely known, the stabilizing feedback law for a is hard to determine. Based on the aforementioned analysis, we set out to design a feedback law for ${\mathit{\tau }}_c$, i.e., ${\mathit{\tau }}_c=\mathcal{T}(\mathit{a})$, by comparing (1) and (12), and making sure that the FNS-induced torque, ${\mathit{\tau }}_m$, converges to ${\mathit{\tau }}_c$. Even if the difference between ${\mathit{\tau }}_m$ and ${\mathit{\tau }}_c$ cannot be fully compensated, the robustness of our controller must ensure the stability of the system in the presence of any disturbances. To facilitate the design of such a controller, Eq. (12) can be rewritten as

$$M(\mathit{q})\ddot{\mathit{q}}+C_m(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+G(\mathit{q})+F(\mathit{q},\dot{\mathit{q}})+\mathit{d}={\mathit{\tau }}_m$$ (13)

which can be further written as

$$M(\mathit{q})\ddot{\mathit{q}}+C_m(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+G(\mathit{q})+F(\mathit{q},\dot{\mathit{q}})+\mathit{\omega }={\mathit{\tau }}_c$$ (14)

where $\mathit{\omega }=\mathit{d}+{\mathit{\tau }}_d$.

To better analyze the controller, we define the trajectory tracking error terms as

$$\mathit{e}\triangleq {\mathit{q}}_d-\mathit{q}$$ (15)

and the sliding mode surface as

$$\mathit{s}\triangleq \dot{\mathit{e}}+\alpha \mathit{e}$$ (16)

where ${\mathit{q}}_d$ denotes the reference joint angle vector.

We design two types of robust controllers, i.e., a high compensation controller and an adaptive controller. To compare the performance of the two controllers, we will test them individually. The integrated control loop is shown in Fig. 5.

Fig. 5.

The integrated control loop

3.1 High Compensation Controller Design.

Consider a controller design that can generate ${\mathit{\tau }}_c$ as

$$\begin{array}{c}{\mathit{\tau }}_c=\widehat{M}({\ddot{\mathit{q}}}_d+\lambda {\dot{\mathit{q}}}_d-\lambda \dot{\mathit{q}}+{\kappa }_2\mathit{s})+{\widehat{C}}_m\mathit{s}\\ +\widehat{C}+\widehat{F}+\widehat{G}+{\kappa }_0\mathit{s}+{\kappa }_1\text{sgn}(\mathit{s})\\ +{\zeta }_0||\mathit{e}|{|}^2\text{sgn}(\mathit{s})+{\zeta }_1||\dot{\mathit{e}}|{|}^2\text{sgn}(\mathit{s})+\Gamma \text{sgn}({\dot{\mathit{q}}}^T\mathit{s})\dot{\mathit{q}}\end{array}$$ (17)

where ${\kappa }_0\in {ℝ}^{+},\hspace{0.17em}{\kappa }_1\in {ℝ}^{+}$, ${\kappa }_2\in {ℝ}^{+}$ are control gains, $\Gamma \in {ℝ}^{3\times 3}$ is a positive definite control gain matrix, $\widehat{M}\in {ℝ}^{3\times 3}$, ${\widehat{C}}_m\in {ℝ}^{3\times 3},\hspace{0.17em}\widehat{C}\in {ℝ}^3$, $\widehat{F}\in {ℝ}^3,\hspace{0.17em}\widehat{G}\in {ℝ}^3$, are estimates of M, C_m, C, F, G, respectively, where $C(\mathit{q},\dot{\mathit{q}})=C_m(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}$ [38].

Let's define a Lyapunov candidate, $V\in ℝ$

$$V=\frac{1}{2}{\mathit{s}}^{T}M\mathit{s}$$ (18)

such that ${\lambda }_m||\mathit{s}|{|}^2\le V\le {\lambda }_M||\mathit{s}|{|}^2$, where λ_m and λ_M are the minimum and maximum eigenvalue of M, respectively. The time derivative of V is

$$\dot{V}={\mathit{s}}^{T}M\dot{\mathit{s}}+\frac{1}{2}{\mathit{s}}^T\dot{M}\mathit{s}$$ (19)

On substituting Eq. (17) into Eq. (19), also with appropriate gain selection, i.e., κ₀, κ₁, κ₂ and each element of Γ being sufficiently high, we can obtain

$$\dot{V}<-{\kappa }_2{\mathit{s}}^{T}M\mathit{s}=-2{\kappa }_{2}V$$

We can solve that

$$V(t)<V(0)\text{exp}(-2{\kappa }_{2}t)$$

Therefore, V(t) exponentially approaches zero with time, which means that $\mathit{s}(t)$ exponentially approaches zero with time. Therefore, it proves the stability.

3.2 Adaptive Controller Design.

Although the high compensation controller can achieve an exponentially stable tracking, the sign functions in the controller usually cause chattering since real control system does not possess perfect switching [41]. To alleviate this disadvantage, we can use adaptive terms to replace the sign functions. The key of this control method is to approximate the structured and unstructured disturbance terms with an adaptive term, i.e.,

$$\begin{array}{c}\tilde{M}(\lambda {\dot{\mathit{q}}}_d+{\ddot{\mathit{q}}}_d)+\tilde{C}+\tilde{F}+\tilde{G}-\lambda \tilde{M}\dot{\mathit{q}}+\mathit{\omega }\\ =Y(\mathit{s},{\mathit{s}}_d)\mathit{\theta }+\mathit{\upsilon }(\mathit{q},\dot{\mathit{q}})\end{array}$$ (20)

where $Y(·,·)\in {ℝ}^{3\times p}$ denotes a regression function, $\mathit{\theta }\in {ℝ}^p$ is the ideal parameter vector, ${\mathit{s}}_d=\lambda {\dot{\mathit{q}}}_d+{\ddot{\mathit{q}}}_d$, and $||\mathit{\upsilon }(\mathit{q},\dot{\mathit{q}})||\le \overline{\upsilon }$.

The controller can be designed as

$$\begin{array}{c}{\mathit{\tau }}_c=\widehat{M}\left({\ddot{\mathit{q}}}_d+\lambda {\dot{\mathit{q}}}_d-\lambda \dot{\mathit{q}}+{\kappa }_2\mathit{s}\right)+{\widehat{C}}_m\mathit{s}\\ +\widehat{C}+\widehat{F}+\widehat{G}+{\kappa }_0\mathit{s}+Y\left(\mathit{s},{\mathit{s}}_d\right)\widehat{\mathit{\theta }}+\frac{\mathit{s}}{4{\mathrm{\epsilon }}_a}\end{array}$$ (21)

where $\widehat{\mathit{\theta }}\in {ℝ}^p$ denotes the estimation of $\mathit{\theta },\hspace{0.17em}{\mathrm{\epsilon }}_a\in {ℝ}^{+}$ is a control gain.

Lyapunov function can be designed as

$$V=\frac{1}{2}{\mathit{s}}^{T}M\mathit{s}+\frac{1}{2}{\tilde{\mathit{\theta }}}^T{\Omega }^{-1}\tilde{\mathit{\theta }}$$ (22)

where $\tilde{\mathit{\theta }}=\mathit{\theta }-\widehat{\mathit{\theta }}$, and ${\Omega }^{-1}\in {ℝ}^{p\times p}$ denotes a full rank positive definite matrix. Its time derivative is

$$\dot{V}={\mathit{s}}^{T}M\dot{\mathit{s}}+\frac{1}{2}{\mathit{s}}^T\dot{M}\mathit{s}+{\tilde{\mathit{\theta }}}^T{\Omega }^{-1}\dot{\tilde{\mathit{\theta }}}$$ (23)

Designing the adapting law as

$$\dot{\widehat{\mathit{\theta }}}=\Omega (-Y^T\mathit{s}+{\gamma }_a\widehat{\mathit{\theta }})$$ (24)

on substituting (21) and (24) into (23) yields

$$V(t)<V(0)\text{exp}(-p_{a}t)+c_a(1-\text{exp}(-p_{a}t))$$

where $p_a=\text{min}\{2{\kappa }_2,2{\gamma }_a\}$ and $c_a=\frac{{\mathrm{\epsilon }}_a||\tilde{\mathit{\theta }}|{|}^2+\frac{{\gamma }_a^2||\mathit{\theta }|{|}^2}{4{\mathrm{\epsilon }}_a}+{\mathrm{\epsilon }}_a{\overline{\upsilon }}^2}{p_a}$. Therefore, for any t, V(t) is bounded by a region, which can be reduced by tuning the gains, i.e., uniformly ultimately bounded (U.U.B.) [42]. The detailed proof can be found in the Appendix.

4 Results

4.1 Simulation Setup.

In testing the control framework, a more anatomically realistic musculoskeletal model of the trunk was created with the opensim musculoskeletal modeling software. In the opensim, not only the trunk and pelvic but also the arms, hands, and head were incorporated to enhance the reality of the simulation with respect to application in a real person with SCI. The opensim model accounts for realistic neuromuscular dynamics of the muscles as well. The overall neuromusculoskeletal model is shown in Fig. 5. In the controller, the model parameters, $\widehat{M},\hspace{0.17em}\widehat{C},\hspace{0.17em}\widehat{G}$, were estimated by deriving the two-link inverted pendulum dynamics with three degree-of-freedom (see Fig. 3(b)), and $\widehat{F}$ is neglected so that $\tilde{F}=\widehat{F}$. Since the controller model parameters were derived from ideal pendulum dynamics, which may not fully describe the realistic trunk dynamics, the difference is accommodated by disturbance term in Eq. (1). As long as the disturbances are bounded, as what is stated in Assumption 1, the controller can reject them during the control process.

The estimated mapping, $\widehat{\mathcal{M}}(\mathit{\tau })$ in Eq. (3), is formed by a SONN, which is a 20 × 20 (400) Kohonen-type NN. The mapping $\widehat{\mathcal{M}}(\mathit{\tau })$ was trained with extensive data obtained by simulations with the musculoskeletal model used for the control process in which the trunk traverses several trajectories to and away from the erect posture. The generated trajectories were input to the opensim inverse dynamics engine to compute the required joint torques. Thereafter, dynamic optimization was performed with the single-objective genetic algorithm module of the dakota package [43] to solve the muscle force-distribution (i.e., redundancy) problem [35,36] for the equivalent muscle activations. The function, ${\widehat{f}}_N$, is formed by CoNN, and the input $\mathit{x}={[\mathit{1},{\mathit{\tau }}_c,{\mathit{\tau }}_d]}^T$ and ${\sigma }_v(·)$ and ${\sigma }_w(·)$ are set to be tanh and sigmoid, respectively. The feedforward fashion of SONN and the simple structure of the CoNN enable fast computation and parameter updating, which supports high control frequency.

The opensim neuromusculoskeletal model and the SONN were compiled into an S-function for ease of simulation in matlab/simulink (The Mathworks Inc., Natick, MA). In the simulation, the reference postures were set to initially result in flexion of the lumbar and pelvic joints, and this position was maintained while lateral bending was initiated to both the left and right. Finally, the trunk returned to the original erect posture. During the control process, the range of pelvic pitch angle was from 0 to 10, lumbar pitch was 0–15, and lumbar roll was –10 to 10. Perturbations, i.e., 10 degrees added to the lumbar pitch and 5 deg lumbar roll, were also applied to the lumbar pitch and lumbar roll at 5.5 s and 8 s, respectively, which was to mimic the scenario that a person is pushed or nudged or is picking up an object with the trunk at a pose that is away from an erect posture.

4.2 Performance Analysis.

Except for the control accuracy (in terms of RMSE), we also evaluated the torque integration index (T.I.I.), i.e., $\mathit{TII}=\int |\mathit{\tau }(t)|dt$, and activation integration index (A.I.I.), i.e., $\mathit{AII}=\int \mathit{a}(t)dt$, which characterize the energy consumption. We also evaluated the torque jerk index (T.J.I.), i.e., $\mathit{TJI}=\int |\dot{\mathit{\tau }}(t)|dt$, which characterize the chattering level.

4.3 Simulation Results.

The two derived controllers (i.e., high compensation controller (high comp.) and adaptive controller (adapt.)) were applied to control the defined motion. The results are depicted in Figs. 6 and 7. Fig. 6 shows the control results without perturbation while Fig. 7 shows the response of the system to perturbations as described. The posture changes corresponding to Fig. 6 are shown in Fig. 8, which depicts the visualized comparison between the two methods in terms of control accuracy. The inputs are shown in Figs. 9 and 10. The proposed control methods were able to drive the actual joint angles to the reference even in the presence of the modeling disturbances and unexpected perturbations.

Fig. 6.

The figure shows the reference (ref) trajectory tracking performance, where the motion driven by the high compensation (high comp.) controller and the motion driven by the adaptive (adapt.) controller are plotted. Angular trajectories of the trunk joints without perturbation.

Fig. 7.

Angular trajectories of the trunk joints with perturbation which were added to lumbar pitch at 5.5 s and lumbar roll at 8.5 s

Fig. 8.

This figure shows the posture changes corresponding to Fig. 6. The left model depicts the motion driven by the high compensation (high comp.) controller while the right is the adaptive (adapt.) controller.

Fig. 9.

The figure shows the torques determined by the controller and the torques caused by the FNS

Fig. 10.

The figures show the FNS-caused muscle activation. The muscles were divided into four groups, i.e., group 1: GM 1-3 and AM, group 2: SM, SA and IL, group 3: ES, RA and PS, group 4: EO, IO and QL. The right (R) and left (L) were also separated for the evaluation.

We compared our method with two existing control methods, i.e., proportional–integral–derivative (PID) controller [44] and PD-DC [45], in terms of RMSE, T.I.I., A.I.I., and T.J.I. We chose the two methods because they can be blindly used without stability analysis. PID is proven an effective control method with many applications. PD-DC can stabilize uncertain Euler–Lagrange system with high gains while the Lyapunov–Krasovskii functional can compensate for the neuromuscular response delay. In this paper, we replace the robust controller with the two existing methods to investigate the control performance. We also compared the results with a hybrid controller (hybrid) combined of the high comp. and adapt. (Figs. 9 and 11)

$$\begin{array}{c}{\mathit{\tau }}_c=\widehat{M}\left({\ddot{\mathit{q}}}_d+\lambda {\dot{\mathit{q}}}_d-\lambda \dot{\mathit{q}}+{\kappa }_2\mathit{s}\right)+{\widehat{C}}_m\mathit{s}\\ +\widehat{C}+\widehat{F}+\widehat{G}+{\kappa }_0\mathit{s}+{\kappa }_1\text{sgn}\left(\mathit{s}\right)\\ +Y\left(\mathit{s},{\mathit{s}}_d\right)\widehat{\mathit{\theta }}+\frac{\mathit{s}}{4{\mathrm{\epsilon }}_a}\end{array}$$ (25)

Fig. 11.

This figure shows the A.I.I. (average of R and L) of the muscles under different controllers

where a sign function, ${\kappa }_1\text{sgn}(\mathit{s})$, is added to the adapt. to overcome the unstructured disturbance.

We tried the best to tune the gain to obtain the best results. The results are summarized in Table 1. As expected, High Comp. shows a more powerful effect in terms of control accuracy. However, its FNS-induced torques chatter severely. The adapt., on the other hand, provides less saturated and smoother inputs. But its control performance is compromised. It can also be seen that, in the adapt. case, the joint angles are being regulated to the desired positions gradually during 2 s to 8 s due to the adaption. The high comp., however, regulated the joint angles very quickly. The hybrid can be more robust than the adapt. and less chattering in the inputs than the high comp. method. In practice, different controllers can be selected based on different applications.

Table 1.

The table summarizes the RMSE, T.I.I., and T.J.I. of the proposed controllers, i.e., high comp. (HC.), adapt. and the hybrid, and the existing controllers, i.e., PID and PD-DC

	HC.	Adapt.	Hybrid	PID	PD-DC
Joint	RMSE
Pelvic pitch	1.79	2.97	2.57	3.33	29.66
Lumbar pitch	0.50	7.64	0.53	7.75	14.60
Lumbar roll	0.89	1.41	0.78	33.63	12.17
Ave.	1.06	4.00	1.29	14.90	18.81
	T.I.I.(103)/T.J.I.(106)
Pelvic pitch	1.9/6.89	0.6/1.11	1.3/4.99	4.9/2.82	8.8/0.25
Lumbar pitch	5.9/21.50	1.1/2.09	3.9/13.70	5.5/2.16	6.1/0.37
Lumbar roll	0.8/1.32	0.3/1.33	1.1/3.07	3.0/0.05	1.1/0.09
Ave.	2.8/9.90	0.6/1.51	2.1/7.25	4.4/1.67	5.3/0.23

Our framework with the specifically designed controller has been shown superior to the existing methods. This is, perhaps, because our controllers contain the forward compensation terms derived based on the two-link inverted pendulum, which can approximate the trunk skeletal (rigid-body) dynamics. Even though the PD-DC controller produces the lowest average chattering, its overall performance cannot match the proposed controllers. It could be that the PD-DC controller is not able to respond fast to provide control signal so that the chattering is less but the RMSE is high.

All in all, this result implies the potential viability of the proposed control method for implementation in actual human subjects with SCI. Further research will explore the potential computational effort required to implement the controller in real life.

5 Conclusion and Discussion

This paper proposed a robust control framework, which consists of a robust control module, i.e., high comp. or adapt., and an adjustable NN with an anatomy-based updating law, to drive the human trunk motion to follow a reference trajectory and to maintain it in a stable manner. The proposed control method was derived to be free of any form of system-identification since the skeletal and neuromuscular dynamics are not completely known. The results suggest that the control task was satisfactorily achieved while unexpected perturbations can also be rejected due to the robustness of the controller. Inevitably, limitations exist in the paper. The joint angle and torque are assumed to be perfectly known. However, they can only be estimated in practice. The resulting torque caused by the FNS with adapt. is much smoother than high comp., the stimulation itself still shows a significant chattering. In this work, muscle fatigue was not considered so that the muscle response to the stimulation is invariant to the duration of the stimulation. In our future work, this control framework will be extended by more sophisticated techniques, e.g., joint angle and torque estimation, controller that can provide smoother input signal, and intelligent control methods that can provide stabilizing signals while consider muscle fatigue. Nevertheless, the proposed control framework provides a concrete preliminary demonstration for the human-involved experiments. It will be tested on human subjects to examine its clinical usage.

Appendix

A.1 High Compensation Stability

Differentiating (16) with respect to time and substituting (13) and (15), we have

$$\begin{array}{c}\dot{\mathit{s}}={\ddot{\mathit{q}}}_d+\lambda {\dot{\mathit{q}}}_d-\lambda \dot{\mathit{q}}+M^{-1}C+M^{-1}F+M^{-1}G\\ +M^{-1}\mathit{\omega }-M^{-1}{\mathit{\tau }}_m\end{array}$$ (A1)

After multiplying (A1) by M(q), we have

$$\begin{array}{c}M\dot{\mathit{s}}=M{\ddot{\mathit{q}}}_d+\lambda M{\dot{\mathit{q}}}_d-\lambda M\dot{\mathit{q}}\\ +C+F+G+\mathit{\omega }-{\mathit{\tau }}_m\end{array}$$

and on substituting (17), the following equation is obtained:

$$\begin{array}{c}M\dot{\mathit{s}}=M{\ddot{\mathit{q}}}_d+\lambda M{\dot{\mathit{q}}}_d-\lambda M\dot{\mathit{q}}+C+F+G-\widehat{M}{\ddot{\mathit{q}}}_d\\ -\widehat{C}-\widehat{F}-\widehat{G}-{\widehat{C}}_m\mathit{s}-\widehat{M}(\lambda {\dot{\mathit{q}}}_d-\lambda \dot{\mathit{q}}+{\kappa }_2\mathit{s})\\ -{\kappa }_1\text{sgn}(\mathit{s})-{\kappa }_0\mathit{s}-{\zeta }_0||\mathit{e}|{|}^2\text{sgn}(\mathit{s})-{\zeta }_1||\dot{\mathit{e}}|{|}^2\text{sgn}(\mathit{s})\\ -\Gamma \text{sgn}({\dot{\mathit{q}}}^T\mathit{s})\dot{\mathit{q}}+\mathit{\omega }\end{array}$$

which can be reduced to

$$\begin{array}{c}M\dot{\mathit{s}}=-C_m\mathit{s}+\tilde{M}({\ddot{\mathit{q}}}_d+\lambda {\dot{\mathit{q}}}_d)+\tilde{C}+\tilde{F}+\tilde{G}\\ -{\kappa }_1\text{sgn}(\mathit{s})+{\kappa }_2\tilde{M}\mathit{s}+{\tilde{C}}_m\mathit{s}-{\kappa }_0\mathit{s}\\ -{\zeta }_0||\mathit{e}|{|}^2\text{sgn}(\mathit{s})-{\zeta }_1||\dot{\mathit{e}}|{|}^2\text{sgn}(\mathit{s})-\Gamma \text{sgn}({\dot{\mathit{q}}}^T\mathit{s})\dot{\mathit{q}}\\ -\lambda \tilde{M}\dot{\mathit{q}}-{\kappa }_{2}M\mathit{s}+\mathit{\omega }\end{array}$$

where $\tilde{M}=M-\widehat{M}={\{{\tilde{m}}_{ij}\}}_{i,j=1,2,3}$, ${\tilde{C}}_m=C_m-{\widehat{C}}_m,\hspace{0.17em}\tilde{C}=C-\widehat{C},\hspace{0.17em}\tilde{F}=F-\widehat{F}$, $\tilde{G}=G-\widehat{G}$. This equation can be further simplified to

$$\begin{array}{c}M\dot{\mathit{s}}=-C_m\mathit{s}+\tilde{\mathit{n}}+\tilde{\mathit{y}}-{\kappa }_1\text{sgn}(\mathit{s})-\lambda \tilde{M}\dot{\mathit{q}}\\ -\Gamma \text{sgn}({\dot{\mathit{q}}}^T\mathit{s})\dot{\mathit{q}}+\Upsilon \mathit{s}-{\kappa }_0\mathit{s}-{\kappa }_{2}M\mathit{s}\\ -{\zeta }_0||\mathit{e}|{|}^2\text{sgn}(\mathit{s})-{\zeta }_1||\dot{\mathit{e}}|{|}^2\text{sgn}(\mathit{s})\end{array}$$ (A2)

In Eq. (A2) $\Upsilon \in {ℝ}^{3\times 3}$ is defined as

$$\Upsilon \triangleq \left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& {\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}& {\epsilon }_{23}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}\end{array}\right]=\underset{(\mathit{q},\dot{\mathit{q}})\in {\mathcal{Q}}_1}{\text{sup}}(|{\kappa }_2\tilde{M}+{\tilde{C}}_m|)$$

where ${\epsilon }_{ij}\in ℝ\hspace{0.17em}\forall i,j=1,2,3$ denotes the maximal absolute estimating error of each element [46] of $|{\kappa }_2\tilde{M}+{\tilde{C}}_m|,\hspace{0.17em}\tilde{\mathit{y}}\in {ℝ}^3$ is defined as $\tilde{\mathit{y}}=\tilde{C}+\tilde{F}+\tilde{G}$, $\tilde{\mathit{n}}\in {ℝ}^3$ is defined as $\tilde{\mathit{n}}\triangleq {[{\tilde{n}}_1,{\tilde{n}}_2,{\tilde{n}}_3]}^T=\tilde{M}{\stackrel{\dot{}\dot{}}{\mathit{q}}}_d+\lambda \tilde{M}{\dot{\mathit{q}}}_d+\mathit{\omega }$. It is also assumed that $\tilde{\mathit{y}}$ is bounded as [38]

$$||\tilde{\mathit{y}}||\le \widetilde{Y_h}(||\mathit{e},\hspace{0.17em}\dot{\mathit{e}}|{|}^2)\le y_0||\mathit{e}|{|}^2+y_1||\dot{\mathit{e}}|{|}^2$$ (A3)

where ${\tilde{Y}}_h$ is a positive and monotonically increasing function, and thus ${\eta }_0,\hspace{0.17em}{\eta }_1\in {ℝ}^{+}$.

On substituting Eq. (A2) into Eq. (19), and by applying Property 2, we obtain

$$\begin{array}{c}\dot{V}=-{\kappa }_1{\mathit{s}}^T\text{sgn}(\mathit{s})+{\mathit{s}}^T\tilde{\mathit{n}}+{\mathit{s}}^T\tilde{\mathit{y}}+{\mathit{s}}^T\Upsilon \mathit{s}-{\mathit{s}}^T{\kappa }_0\mathit{s}\\ -\lambda {\mathit{s}}^T\tilde{M}\dot{\mathit{q}}-{\mathit{s}}^T\Gamma \text{sgn}({\dot{\mathit{q}}}^T\mathit{s})\dot{\mathit{q}}-{\mathit{s}}^T{\kappa }_{2}M\mathit{s}\\ \le -{\kappa }_1{\mathit{s}}^T\text{sgn}(\mathit{s})+|s_1||{\tilde{n}}_1|+|s_2||{\tilde{n}}_2|+|s_3||{\tilde{n}}_3|\\ +{\mathit{s}}^T(y_0||\mathit{e}|{|}^2+y_1||\dot{\mathit{e}}|{|}^2)+{\mathit{s}}^T\Upsilon \mathit{s}-{\mathit{s}}^T{\kappa }_0\mathit{s}\\ -{\zeta }_0||\mathit{e}|{|}^2{\mathit{s}}^T\text{sgn}(\mathit{s})-{\zeta }_1||\dot{\mathit{e}}|{|}^2{\mathit{s}}^T\text{sgn}(\mathit{s})-\lambda {\mathit{s}}^T\tilde{M}\dot{\mathit{q}}\\ -{\mathit{s}}^T\Gamma \text{sgn}({\dot{\mathit{q}}}^T\mathit{s})\dot{\mathit{q}}-{\kappa }_2{\mathit{s}}^{T}M\mathit{s}\end{array}$$ (A4)

To make Eq. (A4) negative definite, we design

$$\begin{array}{c}{\kappa }_0>\text{max}\{|{\epsilon }_i|;\forall i,j=1,2,3\}\\ {\kappa }_1>\text{max}\{|{\tilde{n}}_1|,|{\tilde{n}}_2|,|{\tilde{n}}_3|\}\\ {\zeta }_0>y_0\\ {\zeta }_1>y_1\\ \Gamma =[\begin{array}{c}{\gamma }_{ij}\end{array}]\hspace{0.17em}\forall i,j=1,2,3\hspace{0.17em}{\gamma }_{ij}\ge \lambda {\tilde{m}}_{ij}\end{array}$$ (A5)

By applying the control gain conditions (A5), (A4) can be further simplified to

$$\dot{V}<-{\kappa }_2{\mathit{s}}^{T}M\mathit{s}=-2{\kappa }_{2}V$$

We can solve that

$$V(t)<V(0)\text{exp}(-2{\kappa }_{2}t)$$

Therefore, the stability is proven.

A.2 Adaptive Stability

Define two quantities

$$\begin{array}{c}{\tilde{\mathit{\mu }}}_0=\tilde{C}+\tilde{F}+\tilde{G}-\lambda \tilde{M}\dot{\mathit{q}}+\mathit{\omega }\\ {\tilde{\mathit{\mu }}}_1=& \tilde{M}(\lambda {\dot{\mathit{q}}}_d+{\ddot{\mathit{q}}}_d)\end{array}$$

which can be expressed as

$$\begin{array}{c}{\tilde{\mathit{\mu }}}_0=Y_0(\mathit{q},\dot{\mathit{q}}){\mathit{\theta }}_0+{\mathit{\upsilon }}_0(\mathit{q},\dot{\mathit{q}})\\ {\tilde{\mathit{\mu }}}_1=Y_1(\mathit{q},\dot{\mathit{q}}){\mathit{\theta }}_1°{\mathit{s}}_d=I_{3\times 3}{\mathit{s}}_d{Y}_1(\mathit{q},\dot{\mathit{q}}){\mathit{\theta }}_1=Y_{1d}(\mathit{s},{\mathit{s}}_d){\mathit{\theta }}_1\end{array}$$

where $Y_0(\mathit{q},\dot{\mathit{q}})\in {ℝ}^{3\times p_0}$ and $Y_1(\mathit{q},\dot{\mathit{q}})\in {ℝ}^{3\times p_1}$ ( $p_0+p_1=p,\hspace{0.17em}{p}_0,p_1>3$,) are regression functions, $\mathit{\upsilon }(\mathit{q})\in {ℝ}^{3\times p_0}$ is a bias vector, ${\mathit{\theta }}_0\in {ℝ}^{p_0}$ and ${\mathit{\theta }}_1\in {ℝ}^{p_1}$ are adaptive parameters, $I_{3\times 3}$ is an three-by-three identity matrix, and $Y_{1d}(\mathit{q},{\mathit{q}}_d)=I_{3\times 3}{\mathit{s}}_d{Y}_1(\mathit{q},\dot{\mathit{q}})$.

Write a more compact form

$$\tilde{\mathit{\mu }}=[\begin{array}{cc}{Y}_1(\mathit{q},\dot{\mathit{q}})& Y_{1d}(\mathit{s},{\mathit{s}}_d)\end{array}]\left[\begin{array}{c}{\mathit{\theta }}_0\\ {\mathit{\theta }}_1\end{array}\right]+\left[\begin{array}{c}{\mathit{\upsilon }}_0(\mathit{q},\dot{\mathit{q}})\\ \mathbf{0}\end{array}\right]$$

which is

$$\tilde{\mathit{\mu }}=Y(\mathit{s},{\mathit{s}}_d)\mathit{\theta }+\mathit{\upsilon }(\mathit{q},\dot{\mathit{q}})$$ (A6)

where $Y=[\begin{array}{cc}{Y}_1& Y_{1d}\end{array}],\hspace{0.17em}\mathit{\theta }={[\begin{array}{cc}{\mathit{\theta }}_0& {\mathit{\theta }}_1\end{array}]}^T,\hspace{0.17em}\mathit{\upsilon }=[\begin{array}{cc}{\mathit{\upsilon }}_0& \mathit{0}\end{array}]$, which coincides (20).

The approximation of Eq. (A6) can be written as

$$\widehat{\tilde{\mathit{\mu }}}=Y(\mathit{s},{\mathit{s}}_d)\widehat{\mathit{\theta }}$$ (A7)

so that we can obtain

$$\tilde{\mathit{\mu }}-\widehat{\tilde{\mathit{\mu }}}=Y(\mathit{s},{\mathit{s}}_d)\tilde{\mathit{\theta }}+\mathit{\upsilon }(\mathit{q},\dot{\mathit{q}})$$ (A8)

Combining Eq. (21), (A1), and (A8), we obtain

$$\begin{array}{c}M\dot{\mathit{s}}=-C_m\mathit{s}+{\kappa }_2\tilde{M}\mathit{s}+{\tilde{C}}_m\mathit{s}-{\kappa }_0\mathit{s}\\ -{\kappa }_{2}M\mathit{s}+Y\tilde{\mathit{\theta }}+\mathit{\upsilon }-\frac{\mathit{s}}{4{\mathrm{\epsilon }}_a}\end{array}$$ (A9)

which can be substituted into Eq. (23) to obtain

$$\begin{array}{c}\dot{V}=-{\kappa }_0{\mathit{s}}^T\mathit{s}+{\mathit{s}}^T\Upsilon \mathit{s}-{\kappa }_2{\mathit{s}}^{T}M\mathit{s}\\ -\frac{{\mathit{s}}^T\mathit{s}}{4{\mathrm{\epsilon }}_a}+{\mathit{s}}^{T}Y\tilde{\mathit{\theta }}+{\mathit{s}}^T\mathit{\upsilon }+{\tilde{\mathit{\theta }}}^T{\Omega }^{-1}\dot{\tilde{\mathit{\theta }}}\end{array}$$ (A10)

Also, Eq. (24) leads to

$$\dot{\tilde{\mathit{\theta }}}=\Omega (-Y^T\mathit{s}+{\gamma }_a\widehat{\mathit{\theta }})$$ (A11)

which can be substituted to Eq. (A10) to obtain

$$\begin{array}{c}\dot{V}=-{\kappa }_0{\mathit{s}}^T\mathit{s}+{\mathit{s}}^T\Upsilon \mathit{s}-{\kappa }_2{\mathit{s}}^{T}M\mathit{s}\\ -\frac{{\mathit{s}}^T\mathit{s}}{4{\mathrm{\epsilon }}_a}+{\mathit{s}}^T\mathit{\upsilon }+{\mathit{s}}^{T}Y\tilde{\mathit{\theta }}+{\tilde{\mathit{\theta }}}^T\left(-Y^T\mathit{s}+{\gamma }_a\widehat{\mathit{\theta }}\right)\end{array}$$

The gain selection criteria for κ₀ keeps from the high compensation method, we have

$$\dot{V}\le -{\kappa }_2{\mathit{s}}^{T}M\mathit{s}-\frac{{\mathit{s}}^T\mathit{s}}{4{\mathrm{\epsilon }}_a}+{\mathit{s}}^T\mathit{\upsilon }+{\gamma }_a{\tilde{\mathit{\theta }}}^T\mathit{\theta }-{\gamma }_a{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}$$ (A12)

Applying Young's inequality

$$\begin{array}{c}{\mathit{s}}^T\mathit{\upsilon }\le \frac{||\mathit{s}|{|}^2}{4{\mathrm{\epsilon }}_a}+{\mathrm{\epsilon }}_a||\mathit{\upsilon }|{|}^2\\ {\gamma }_a{\tilde{\mathit{\theta }}}^T\mathit{\theta }\le \frac{{\gamma }_a^2||\mathit{\theta }|{|}^2}{4{\mathrm{\epsilon }}_a}+{\mathrm{\epsilon }}_a||\tilde{\mathit{\theta }}|{|}^2\end{array}$$

we can rewrite Eq. (A12) to

$$\begin{array}{c}\dot{V}\le -{\kappa }_2{\mathit{s}}^{T}M\mathit{s}-{\gamma }_a{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}+{\mathrm{\epsilon }}_a||\tilde{\mathit{\theta }}|{|}^2+\frac{{\gamma }_a^2||\mathit{\theta }|{|}^2}{4{\mathrm{\epsilon }}_a}+{\mathrm{\epsilon }}_a{\overline{\upsilon }}^2\\ \le -p_{a}V+c_a\end{array}$$ (A13)

By defining $p_a=\text{min}\{2{\kappa }_2,2{\gamma }_a\}$ and $c_a=\frac{{\mathrm{\epsilon }}_a||\tilde{\mathit{\theta }}|{|}^2+\frac{{\gamma }_a^2||\mathit{\theta }|{|}^2}{4{\mathrm{\epsilon }}_a}+{\mathrm{\epsilon }}_a{\overline{\upsilon }}^2}{p_a}$, we can obtain

$$V(t)<V(0)\text{exp}(-p_{a}t)+c_a(1-\text{exp}(-p_{a}t))$$

Therefore, the proof is complete.

Funding Data

• National Institutes of Health (Grant No. 1R01NS101043-01; Funder ID: 10.13039/100000002).

• Department of Defense, SCIR Program (Grant No. W81XWH-17-1-0240; Funder ID: 10.13039/100006370).