# High-Order Sliding Mode-Based Synchronous Control of a Novel Stair-Climbing Wheelchair Robot

Authors: Juanxiu Liu; Yifei Wu; Jian Guo; Qingwei Chen
Journal: Journal of Control Science and Engineering
 (2015)

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## Abstract

For the attitude control of a novel stair-climbing wheelchair with inertial uncertainties and external disturbance torques, a new synchronous control method is proposed via combing high-order sliding mode control techniques with cross-coupling techniques. For this purpose, a proper controller is designed, which can improve the performance of the system under conditions of uncertainties and torque perturbations and also can guarantee the synchronization of the system. Firstly, a robust high-order sliding mode control law is designed to track the desired position trajectories effectively. Secondly, considering the coordination of the multiple joints, a high-order sliding mode synchronization controller is designed to reduce the synchronization errors and tracking errors based on the controller designed previously. Stability of the closed-loop system is proved by Lyapunov theory. The simulation is performed by MATLAB to verify the effectiveness of the proposed controller. By comparing the simulation results of two controllers, it is obvious that the proposed scheme has better performance and stronger robustness.


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## Body

## 1. Introduction

With the rapid increase of the elderly population over the age of 60, population aging will be an outstanding performance of global population trends in the 21st century. Aging society brings a lot of problems, such as nursing for the elderly and medical problems. At the same time, thousands of people lost the ability to walk each year caused by a variety of accidents, natural disasters, and diseases. With the development of the society and the improvement of human civilization, the people with disabilities need to use modern high-tech to improve their freedom and quality of life. Hence, wheelchair robot used to help the disabled or elderly people walking has become a hot research area in recent years.Although the barrier-free accessibility has been disseminated in recent years, stairs and other architectural barriers still exist in many cities and buildings. Since the standard wheelchair has no capability of crossing barriers, a number of stair-climbing wheelchairs which can help the disabled or elderly people overcoming obstacles have been researched. The common stair-climbing mechanisms used in the stair-climbing wheelchairs are tracks, wheels, and hybrid structures. Tracked stair-climbing wheelchair can guarantee the stability of the users in the process of ascending and descending stairs. Lawn et al. [1] designed a tracked wheelchair capable of negotiating large number of twisting and irregular stairs. In [2], a wheelchair using cluster wheels was developed. The wheelchair seat was kept stable during the stairs ascending process and the user needed not to face down the stairs. In [3], Wheelchair.q using triple cluster wheels was designed. The cluster wheels systems usually have complex mechanisms and the stability is lower than crawler systems. A stair-climbing robot with legs and wheels was designed in [4]. Chen and Pham [5] designed a prototype which was comprised of a pair of rotational multilimbed structures. There are some other design schemes adopting hybrid structures in [6–8].Since the stair-climbing wheelchair is used in complex terrain, the first thing that should be considered is high precious position control for wheelchair system. In [2], PID control was used to provide appropriate torque during climbing process. A fuzzy controller was applied to correct the errors in direction and position misalignment, so that the final posture of the tracked mobile robot was corrected in [9]. In [10], an active tension control law combined with the computed torque method was obtained for wheelchair robot during the stair-climbing process, which can track the reference input curve of homonymic constraint force when tracking reference input curve of each joint. Although the control strategies mentioned above can make some efforts for the control of stair-climbing wheelchair, various system uncertainties and torque perturbations were not taken into consideration.Since the sliding mode control has strong robustness for system disturbances and unmodeled dynamics, it has been widely used in robot control. Conventional sliding mode control can achieve the first-order sliding motion of the system states, which means the relative degree of the sliding variables is 1. When the system states slide along the preset manifold, only the sliding variable is guaranteed to converge to zero, and its derivative is nonsmooth, so the chattering phenomenon always comes up in the sliding mode control system. In order to solve the chattering which exists in conventional sliding mode control, a variety of methods were proposed such as state-dependent gain method, observer-based chattering suppression, and the high-order sliding model control (HOSMC) in [11, 12]. In [13], with the HOSMC used in attitude control of large-scale spacecraft, the robustness of the system with respect to uncertainties and external disturbances was improved, and the chattering phenomenon was attenuated. In [14], a HOSMC was designed for a flexible link space robotic arm with payload, which exploited the robustness properties of SMC, while also increasing accuracy by reducing chattering effects.Another important problem that should be considered is the coordination of the stair-climbing wheelchair system due to its characteristic of multiple joints. If the stair-climbing wheelchair works in a noncoordination manner, the assembly task will be failed, and a more serious consequence is that the users will be injured. In [15], a new control approach to position synchronization of multiple motion axes was developed, by incorporating cross-coupling technology into adaptive control architecture. A novel robust adaptive terminal sliding mode position synchronized control approach was proposed for the operation of multiple motion axes system and the convergence of position errors and synchronization errors could be guaranteed in [16]. In [17], the synchronous control of a dual linear motor servosystem was developed by a cross-coupled intelligent complementary sliding mode control system; a better control performance and robustness with regard to uncertainties can be achieved.The motivation of the presented work here is to find a proper controller which has strong rejection capacity against external disturbances and robustness to deal with uncertainties and also can guarantee the synchronization of the system. Based on a novel stair-climbing wheelchair robot which was designed in [18], a new control method is proposed in this paper via combining the HOSMC techniques and cross-coupling techniques. The mechanical structure of the stair-climbing robot is described at first, and then its dynamics with uncertainties and perturbations is analyzed. After that, single joint position tracking controller adopting HOSMC techniques is designed to assure the high accuracy tracking under conditions of uncertainties and torque perturbations, and then a synchronous controller based on HOSMC techniques and cross-coupling techniques is developed to reduce the synchronization errors and position errors at the same time. The stability of the closed-loop system is proved by Lyapunov theory. In the last section, simulations are completed under the same condition by using the HOSM controller and the HOSM synchronous controller, respectively. The two results are compared to validate the effectiveness of the proposed method.This paper is organized as follows. Section2 describes mechanical structure of the stair-climbing wheelchair and its dynamic model with uncertainties and perturbations. In Section 3, a HOSM position controller and a HOSM synchronous controller are proposed and the stability of the closed-loop system is analyzed. Section 4 presents the simulation results. Finally, the conclusion is given in Section 5.


## 2. System Description and Modeling

### 2.1. System Description

It seems to be a quite complex problem to design a staircase climbing wheelchair which is adaptable to various terrains. However, this problem can be solved by simply splitting the process of staircase climbing into two different problems: (a) climbing a single step of variable height; (b) providing stability for the entire mechanism while the wheelchair is on the stair [19]. Based on this design idea, a stair-climbing wheelchair is proposed. Figure 1 shows the virtual prototype of the stair-climbing wheelchair. Figure 2 presents the structure of one side of the wheelchair. The parts of climbing mechanism and two-link mechanism are shown in Figure 2. In this system, all joints are revolute joints driven by motors. The climbing mechanisms, the wheels, and the two-link mechanisms are independent of each other.Figure 1
Virtual prototype of wheelchair.Figure 2
Mechanical structure of wheelchair.The climbing mechanism which contains two swing arms (Figure2) is designed to surmount a single step. The system has four such climbing mechanisms: two front and two rear. The front climbing mechanisms are joined to two-link mechanism, and the rear mechanisms are fixed on the frame. When the wheelchair reaches a step, the swing arms which are driven by motors rotate at a constant speed. When the swing arms touch the top of the step, the wheelchair’s weight is supported by them. The front wheels can now be lifted to surmount the step. When the front wheels touch the top of the step, the weight is now transferred to them. The swing arms continue to rotate to their original positions. This process ends, and the system is now ready to climb the next step.The two-link mechanism is designed to ensure the wheelchair seat always stays at the upright position. Because the wheelchair seat is joined to the frame, when the front wheels rise, the frame will rotate. This should be adjusted by link mechanism. Similarly, the rear climbing mechanisms move when the top of staircase is reached, and the link mechanism should accommodate the seat.The task of the sensorial system is to measure the distances between the wheels and the steps and the information about the steps. Proposed placement of stair sensors is shown in Figure3. Two ultrasound sensors are placed on front wheels in horizontal position to detect stair edge. To measure the distances to next step, two ultrasound sensors are placed on the link mechanism in horizontal position. The width of the step can be calculated by comparing the data of two sets of sensors. There are fourteen rotary transformers (one per joint) to ensure each joint position is measurable. When the positions of the joints are known, the height of swing arms relative to the ground can be calculated, which is defined as h1. Then, two ultrasound sensors are placed on front climbing mechanisms in vertical position to measure the height of swing arms relative to the step, which can be defined as h2. So the step height is the height difference between h1 and h2. Finally, two ultrasound sensors are placed on the frame to measure the positions of rear wheels with respect to the stair edge.Figure 3
Proposed stair sensor placement.The process of climbing stairs is shown in Figure4 and achieved by the following steps.(1)
When the front wheels sense a step, the front climbing mechanisms rotate up at a speed defined by the program. At the same time, the two-link mechanisms also rotate for adjusting the attitude of the seat. The front wheels are raised to surmount the step. The wheelchair moves forward until next step is sensed.(2)
Repeat step (1) until the wheelchair surmounts the second step.(3)
The front and rear climbing mechanisms climb synchronously.(4)
When the top of the staircase is reached, the front climbing mechanisms detect no step edge and remain motionless. The rear mechanisms continue to move to the top of the stair.Figure 4
Steps of stairs ascending process.


### 2.2. System Kinematics and Dynamics

In order to analyze the model of wheelchair, some assumptions are made as follows.Assumption 1.
The wheelchair is a rigid body system. The tires are point contact with the ground, and the contact points have no relative sliding. The tires have no elastic deformation.Assumption 2.
The structure of the robot is symmetrical, so the dynamic analysis is based on the structure which is simplified to planar mechanism.Assumption 3.
The position and attitude of wheelchair are not affected by active control. Each joint has one rotational degree-of-freedom, and the joints are affected by active control.Figure5 shows the kinematic scheme of stair-climbing wheelchair. li(i=1,2,…,7) is the length of wheelchair’s each part. R is the radius of wheels, and r is the radius of small wheels. h is the step height. d is the width of the step. d1 is the distance between the front wheels and the step edge. qi(i=1,2,…,7) is the angle of joint i. δ is the original angle of the swing arm. q0, q˙0, and q¨0 denote the angle of the seat with respect to ground, velocity, and acceleration, respectively. pg is the wheelchair’s center of gravity. mi and ji are the mass and moment of inertia of wheelchair’s each part, respectively.Figure 5
Kinematic scheme of stair-climbing wheelchair.The desired trajectories can be determined by the inverse kinematic model according to the kinematic parameters above. Define theqdi(i=0,1,…,7) as desired trajectory of the joint i. For the safety of user, the trajectories must satisfy qd0=q˙d0=q¨d0=0, making sure that the wheelchair seat always stays at the upright position. For simplicity, the inverse kinematic model is given here, and the derivation process is presented in [20]. When the wheelchair climbs the first and second steps, the inverse kinematic model is the following:(1)qd0=0,qd1=arcsin⁡h+r+l4+l6sin⁡-δ-qd3-R-l1-l2sin⁡qd0l3,qd2=π2-qd1,qd3=-δ+vt,qd4=-π+δ+vt,qd5=d+R-d1+l1sin⁡qd0+l21-cos⁡qd0+l31-cos⁡qd0+qd1l3,qd6=-δ,qd7=-π+δ,where v is the set rotational speed of climbing mechanisms. When h, d, and d1 are accurately measured by sensorial system, the desired trajectories can be generated by (1).According to the previous works in [20], the dynamics can be well approximated by the following equation:(2)Mqq¨+Cq,q˙q˙+Gq=τ,where q is a vector of generalized coordinates, q=q0,q1,…,qnT∈R8, and q˙ and q¨ are velocity vector and acceleration vector, respectively. M(q)∈R8×8 is the positive definite symmetric inertial matrix. C(q,q˙)∈R8×8 is the matrix containing Coriolis force and centripetal force. G(q)∈R8 is the gravity vector and τ=[τ0,τ1,…,τ7]∈R8 is torque input vector. The detailed expressions can be found in the Appendix.In practical applications, there are a lot of uncertain factors such as friction and disturbing torque. Considering the uncertainties and the modeling errors, the dynamic equation (2) can be rewritten as follows:(3)M0qq¨+C0q,q˙q˙+G0q=τ+ρt,(4)ρt=dt-ΔMqq¨-ΔCq,q˙q˙-ΔGq,where M0(q), C0(q,q˙), and G0(q) are nominal values of M(q), C(q,q˙), and G(q), respectively. ΔM(q), ΔC(q,q˙), and ΔG(q) are uncertainties of M(q), C(q,q˙), and G(q), respectively. d(t) is torque perturbations vector, and ρ(t)∈R8 denotes the total uncertainties of the dynamics caused by parameter variations and perturbations.Some assumptions for the system parameters and variables are made as follows.Assumption 4.
M
0
(
q
) is a positively definite matrix, and there are known positive constants Mmax and Mmax′ such that the following inequalities hold: M0-1(q)≤Mmax and M˙0-1(q)≤Mmax′.Assumption 5.
There are known positive constantsρmax and ρmax′ satisfying the following inequalities: ρ(t)≤ρmax and ρ˙(t)≤ρmax′.


## 2.1. System Description

It seems to be a quite complex problem to design a staircase climbing wheelchair which is adaptable to various terrains. However, this problem can be solved by simply splitting the process of staircase climbing into two different problems: (a) climbing a single step of variable height; (b) providing stability for the entire mechanism while the wheelchair is on the stair [19]. Based on this design idea, a stair-climbing wheelchair is proposed. Figure 1 shows the virtual prototype of the stair-climbing wheelchair. Figure 2 presents the structure of one side of the wheelchair. The parts of climbing mechanism and two-link mechanism are shown in Figure 2. In this system, all joints are revolute joints driven by motors. The climbing mechanisms, the wheels, and the two-link mechanisms are independent of each other.Figure 1
Virtual prototype of wheelchair.Figure 2
Mechanical structure of wheelchair.The climbing mechanism which contains two swing arms (Figure2) is designed to surmount a single step. The system has four such climbing mechanisms: two front and two rear. The front climbing mechanisms are joined to two-link mechanism, and the rear mechanisms are fixed on the frame. When the wheelchair reaches a step, the swing arms which are driven by motors rotate at a constant speed. When the swing arms touch the top of the step, the wheelchair’s weight is supported by them. The front wheels can now be lifted to surmount the step. When the front wheels touch the top of the step, the weight is now transferred to them. The swing arms continue to rotate to their original positions. This process ends, and the system is now ready to climb the next step.The two-link mechanism is designed to ensure the wheelchair seat always stays at the upright position. Because the wheelchair seat is joined to the frame, when the front wheels rise, the frame will rotate. This should be adjusted by link mechanism. Similarly, the rear climbing mechanisms move when the top of staircase is reached, and the link mechanism should accommodate the seat.The task of the sensorial system is to measure the distances between the wheels and the steps and the information about the steps. Proposed placement of stair sensors is shown in Figure3. Two ultrasound sensors are placed on front wheels in horizontal position to detect stair edge. To measure the distances to next step, two ultrasound sensors are placed on the link mechanism in horizontal position. The width of the step can be calculated by comparing the data of two sets of sensors. There are fourteen rotary transformers (one per joint) to ensure each joint position is measurable. When the positions of the joints are known, the height of swing arms relative to the ground can be calculated, which is defined as h1. Then, two ultrasound sensors are placed on front climbing mechanisms in vertical position to measure the height of swing arms relative to the step, which can be defined as h2. So the step height is the height difference between h1 and h2. Finally, two ultrasound sensors are placed on the frame to measure the positions of rear wheels with respect to the stair edge.Figure 3
Proposed stair sensor placement.The process of climbing stairs is shown in Figure4 and achieved by the following steps.(1)
When the front wheels sense a step, the front climbing mechanisms rotate up at a speed defined by the program. At the same time, the two-link mechanisms also rotate for adjusting the attitude of the seat. The front wheels are raised to surmount the step. The wheelchair moves forward until next step is sensed.(2)
Repeat step (1) until the wheelchair surmounts the second step.(3)
The front and rear climbing mechanisms climb synchronously.(4)
When the top of the staircase is reached, the front climbing mechanisms detect no step edge and remain motionless. The rear mechanisms continue to move to the top of the stair.Figure 4
Steps of stairs ascending process.


## 2.2. System Kinematics and Dynamics

In order to analyze the model of wheelchair, some assumptions are made as follows.Assumption 1.
The wheelchair is a rigid body system. The tires are point contact with the ground, and the contact points have no relative sliding. The tires have no elastic deformation.Assumption 2.
The structure of the robot is symmetrical, so the dynamic analysis is based on the structure which is simplified to planar mechanism.Assumption 3.
The position and attitude of wheelchair are not affected by active control. Each joint has one rotational degree-of-freedom, and the joints are affected by active control.Figure5 shows the kinematic scheme of stair-climbing wheelchair. li(i=1,2,…,7) is the length of wheelchair’s each part. R is the radius of wheels, and r is the radius of small wheels. h is the step height. d is the width of the step. d1 is the distance between the front wheels and the step edge. qi(i=1,2,…,7) is the angle of joint i. δ is the original angle of the swing arm. q0, q˙0, and q¨0 denote the angle of the seat with respect to ground, velocity, and acceleration, respectively. pg is the wheelchair’s center of gravity. mi and ji are the mass and moment of inertia of wheelchair’s each part, respectively.Figure 5
Kinematic scheme of stair-climbing wheelchair.The desired trajectories can be determined by the inverse kinematic model according to the kinematic parameters above. Define theqdi(i=0,1,…,7) as desired trajectory of the joint i. For the safety of user, the trajectories must satisfy qd0=q˙d0=q¨d0=0, making sure that the wheelchair seat always stays at the upright position. For simplicity, the inverse kinematic model is given here, and the derivation process is presented in [20]. When the wheelchair climbs the first and second steps, the inverse kinematic model is the following:(1)qd0=0,qd1=arcsin⁡h+r+l4+l6sin⁡-δ-qd3-R-l1-l2sin⁡qd0l3,qd2=π2-qd1,qd3=-δ+vt,qd4=-π+δ+vt,qd5=d+R-d1+l1sin⁡qd0+l21-cos⁡qd0+l31-cos⁡qd0+qd1l3,qd6=-δ,qd7=-π+δ,where v is the set rotational speed of climbing mechanisms. When h, d, and d1 are accurately measured by sensorial system, the desired trajectories can be generated by (1).According to the previous works in [20], the dynamics can be well approximated by the following equation:(2)Mqq¨+Cq,q˙q˙+Gq=τ,where q is a vector of generalized coordinates, q=q0,q1,…,qnT∈R8, and q˙ and q¨ are velocity vector and acceleration vector, respectively. M(q)∈R8×8 is the positive definite symmetric inertial matrix. C(q,q˙)∈R8×8 is the matrix containing Coriolis force and centripetal force. G(q)∈R8 is the gravity vector and τ=[τ0,τ1,…,τ7]∈R8 is torque input vector. The detailed expressions can be found in the Appendix.In practical applications, there are a lot of uncertain factors such as friction and disturbing torque. Considering the uncertainties and the modeling errors, the dynamic equation (2) can be rewritten as follows:(3)M0qq¨+C0q,q˙q˙+G0q=τ+ρt,(4)ρt=dt-ΔMqq¨-ΔCq,q˙q˙-ΔGq,where M0(q), C0(q,q˙), and G0(q) are nominal values of M(q), C(q,q˙), and G(q), respectively. ΔM(q), ΔC(q,q˙), and ΔG(q) are uncertainties of M(q), C(q,q˙), and G(q), respectively. d(t) is torque perturbations vector, and ρ(t)∈R8 denotes the total uncertainties of the dynamics caused by parameter variations and perturbations.Some assumptions for the system parameters and variables are made as follows.Assumption 4.
M
0
(
q
) is a positively definite matrix, and there are known positive constants Mmax and Mmax′ such that the following inequalities hold: M0-1(q)≤Mmax and M˙0-1(q)≤Mmax′.Assumption 5.
There are known positive constantsρmax and ρmax′ satisfying the following inequalities: ρ(t)≤ρmax and ρ˙(t)≤ρmax′.


## 3. Control System Design and Stability Analysis

The wheelchair kinematics and dynamics have been given in the previous section, which may make the controller design complicated. A robust control law with respect to uncertainties and perturbations is needed in the high accuracy position tracking of the stair-climbing wheelchair system. In this paper, single joint position tracking controller and a synchronous controller are developed, respectively. Stability of the closed-loop system is analyzed by Lyapunov theory. Here is the design process in detail.


### 3.1. HOSM Controller Design

This section considers the position tracking problem for wheelchair joints, the kinematic and dynamic parameters are known, and the desired trajectory is denoted asqdi. The measureable states are joint position qi and the joint velocity q˙i. The control objective is to design robust chattering-free control signal τ, which ensures that qi always tracks the desired trajectory qdi, in spite of the inertial uncertainties and external disturbance torques.The position tracking error of a single joint is defined as(5)ei=qi-qdi.Sets=e1,e2,…,enT∈Rn, in order to achieve HOSM control of sliding variable s, we define σ as (6)σ=s¨+c1s˙+c0s,where c1 and c0 are both positive constants. The HOSM control law for sliding variable s is as follows:(7)τ=ueq+M0quvss,(8)ueq=M0qq¨d+C0q,q˙q˙+G0q+M0q-c1s˙-c0s,(9)u˙vss=-k1σ-k2σσ,where k1 and k2 are positive constants, satisfying k1>0 and k2>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t).Theorem 1.
Considering system (3), the sliding surface defined in (6) is chosen. When the parameters satisfy k1>0 and k2>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t), the position tracking error fulfills the condition that limt→∞⁡s=0, under the control effort of the control law given in (7).Proof.
Choose the following Lyapunov function:(10)V=12σTσ.Taking the time derivative of V, it can be written as(11)V˙=σTσ˙=σTs⃛+c1s¨+c0s˙.Differentiating (5) and taking (3) yield (12)s¨=q¨-q¨d=M0-1qτ+ρt-G0q-C0q,q˙q˙-q¨d.Substituting (7), (8), and (9) into (12), we have(13)s¨=M0-1qM0qq¨d+ρt+M0q-c1s˙-c0s+M0quvss-q¨d=M0-1qρt-c1s˙-c0s+uvss.Differentiating (13), and considering (9), it can be obtained as follows:(14)s⃛=M˙0-1qρt+M0-1qρ˙t-c1s¨-c0s˙+u˙vss=M˙0-1qρt+M0-1qρ˙t-c1s¨-c0s˙-k1σ-k2σσ.Substituting (14) into (11), we get(15)V˙=σTM˙0-1qρt+M0-1qρ˙t-c1s¨-c0s˙-k1σ-k2σσ+c1s¨+c0s˙=σTM˙0-1qρt+σTM0-1qρ˙t-σTk1σ-k2σ≤σTM˙0-1qρt+σTM0-1qρ˙t-σTk1σ-k2σ=σTM˙0-1qρt+M0-1qρ˙t-k2-σTk1σ<-σTk1σ.
Therefore,σTσ˙<0, when σ≠0, the system is asymptotically stable, and limt→∞⁡s=0 also can be assured.From Theorem1, it can be obtained that the tracking error can converge to zero in spite of uncertainties and perturbation. Figure 6 presents the structure of the HOSM position tracking controller without coordinating the motion of each joint. With the information provided by sensorial system, the reference trajectory qdi is obtained by using the inverse kinematic model. The tracking error ei can be obtained by the feedback signal, and then the controller calculates the joint driving moment τ which is applied to the dynamic model for joint control.Figure 6
Structure of the HOSM controller.


### 3.2. HOSM Synchronous Controller Design

Because the position and attitude of the wheelchair are affected by each joint’s motion, the coordination among the joints is needed to be considered besides the position tracking error of each joint. If the motions of climbing mechanisms are asynchronous, the seat will lean in the process of climbing stairs, and it will result in a series of problems such as the system damage and injury accidents. This section considers the synchronization problem of multiple joints system. The control objective is to design robust chattering-free control signalτ′, which ensures that the position tracking error and synchronization error converge to zero, when qdi, qi, and q˙i are known.For a multiple joints system, the position tracking errors must satisfy the following condition in [15]:(16)limt→∞⁡e1=limt→∞⁡e2=⋯=limt→∞⁡en=0.Considering (17), the synchronization errors are defined as(17)ε1=e1-e2+e1,ε2=e2-e3+e2-e1,⋮εi=ei-ei+1+ei-ei-1,⋮εn=en+en-en-1.The expression means that the synchronization errors of the multiple joints system εi are defined as differential position errors among multiple joints system. Setting ε=ε1,ε2,…,εnT∈Rn, function (17) can be written as(18)ε1ε2⋮εn-1εn=2-10⋯0-12-1⋯0⋮⋱⋱⋱⋮0⋯-12-100⋯-12e1e2⋮en-1en=Te,where T is the positive definite matrix.Settings2=ε, in order to achieve HOSM control of sliding variable s2, define σs as(19)σs=s¨2+c1s˙2+c0s2.Substituting (18), there is(20)σs=Ts¨+c1Ts˙+c0Ts.The HOSM control law for sliding variable s2 is designed as(21)u˙vss′=-k3σs-k4σsσs,where k3>0 and k4>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t).The HOSM synchronous control law which is the overall control law is developed as follows:(22)τ′=ueq+M0quvss+M0quvss′.Theorem 2.
Considering system (3), the sliding surface defined in (19) is chosen. When the parameters satisfy k1>0, k2>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t), k4>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t), and k3>0, the position tracking error e and synchronization error ε can converge to zero simultaneously, under the control effort of the control law given in (22).Proof.
Choose the following Lyapunov function:(23)V=12σTσ+12σsTT-1σs.Differentiating V with respect to time yields(24)V˙=σTσ˙+σsTT-1σ˙s.Considering (14), and differentiating (6) and (20), we get(25)σ˙=u˙vss+u˙vss′+M˙0-1qρt+M0-1qρ˙t,σ˙s=Tu˙vss+Tu˙vss′+TM˙0-1qρt+TM0-1qρ˙t.Substituting (25) into (24), we have(26)σTσ˙=σTM˙0-1qρt+M0-1qρ˙t-k1σ-k2σσ-k3σs-k4σsσs=σTM˙0-1qρt+σTM0-1qρ˙t-σTk1σ-k2σ-σTk3σs-σTk4σsσs≤σTM˙0-1qρt+σTM0-1qρ˙t-σTk1σ-k2σ-σTk3Tσ-k4σTTσσs=σTM˙0-1qρt+M0-1qρ˙t-k2-σTk1σ-σTk3Tσ-k4σTTσσs<-σTk1σ-σTk3Tσ-k4σTTσσs,σsTT-1σ˙s=σsTT-1TM˙0-1qρt+M0-1qρ˙t-k1σ-k2σσ-k3σs-k4σsσs=σsTM˙0-1qρt+σsTM0-1qρ˙t-σTTk1σ-σTTk2σσ-σsTk3σs-k4σs≤σsTM˙0-1qρt+M0-1qρ˙t-k4-σTTk1σ-σTTk2σσ-σsTk3σs<-σTTk1σ-σTTk2σσ-σsTk3σs.From (18), T is positive definite matrix, so it can be obtained that σTσ˙<0 and σsTT-1σ˙s<0. It is proved that the system is stable simultaneously, and the convergence of the position tracking error e and synchronization error ε is assured; that is, limt→∞⁡s=0 and limt→∞⁡ε=0. Figure 7 presents the structure of the HOSM synchronous controller. It can be found that the synchronous control signal is added to the HOSMC control signal.Figure 7
Structure of the HOSM synchronous controller.


## 3.1. HOSM Controller Design

This section considers the position tracking problem for wheelchair joints, the kinematic and dynamic parameters are known, and the desired trajectory is denoted asqdi. The measureable states are joint position qi and the joint velocity q˙i. The control objective is to design robust chattering-free control signal τ, which ensures that qi always tracks the desired trajectory qdi, in spite of the inertial uncertainties and external disturbance torques.The position tracking error of a single joint is defined as(5)ei=qi-qdi.Sets=e1,e2,…,enT∈Rn, in order to achieve HOSM control of sliding variable s, we define σ as (6)σ=s¨+c1s˙+c0s,where c1 and c0 are both positive constants. The HOSM control law for sliding variable s is as follows:(7)τ=ueq+M0quvss,(8)ueq=M0qq¨d+C0q,q˙q˙+G0q+M0q-c1s˙-c0s,(9)u˙vss=-k1σ-k2σσ,where k1 and k2 are positive constants, satisfying k1>0 and k2>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t).Theorem 1.
Considering system (3), the sliding surface defined in (6) is chosen. When the parameters satisfy k1>0 and k2>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t), the position tracking error fulfills the condition that limt→∞⁡s=0, under the control effort of the control law given in (7).Proof.
Choose the following Lyapunov function:(10)V=12σTσ.Taking the time derivative of V, it can be written as(11)V˙=σTσ˙=σTs⃛+c1s¨+c0s˙.Differentiating (5) and taking (3) yield (12)s¨=q¨-q¨d=M0-1qτ+ρt-G0q-C0q,q˙q˙-q¨d.Substituting (7), (8), and (9) into (12), we have(13)s¨=M0-1qM0qq¨d+ρt+M0q-c1s˙-c0s+M0quvss-q¨d=M0-1qρt-c1s˙-c0s+uvss.Differentiating (13), and considering (9), it can be obtained as follows:(14)s⃛=M˙0-1qρt+M0-1qρ˙t-c1s¨-c0s˙+u˙vss=M˙0-1qρt+M0-1qρ˙t-c1s¨-c0s˙-k1σ-k2σσ.Substituting (14) into (11), we get(15)V˙=σTM˙0-1qρt+M0-1qρ˙t-c1s¨-c0s˙-k1σ-k2σσ+c1s¨+c0s˙=σTM˙0-1qρt+σTM0-1qρ˙t-σTk1σ-k2σ≤σTM˙0-1qρt+σTM0-1qρ˙t-σTk1σ-k2σ=σTM˙0-1qρt+M0-1qρ˙t-k2-σTk1σ<-σTk1σ.
Therefore,σTσ˙<0, when σ≠0, the system is asymptotically stable, and limt→∞⁡s=0 also can be assured.From Theorem1, it can be obtained that the tracking error can converge to zero in spite of uncertainties and perturbation. Figure 6 presents the structure of the HOSM position tracking controller without coordinating the motion of each joint. With the information provided by sensorial system, the reference trajectory qdi is obtained by using the inverse kinematic model. The tracking error ei can be obtained by the feedback signal, and then the controller calculates the joint driving moment τ which is applied to the dynamic model for joint control.Figure 6
Structure of the HOSM controller.


## 3.2. HOSM Synchronous Controller Design

Because the position and attitude of the wheelchair are affected by each joint’s motion, the coordination among the joints is needed to be considered besides the position tracking error of each joint. If the motions of climbing mechanisms are asynchronous, the seat will lean in the process of climbing stairs, and it will result in a series of problems such as the system damage and injury accidents. This section considers the synchronization problem of multiple joints system. The control objective is to design robust chattering-free control signalτ′, which ensures that the position tracking error and synchronization error converge to zero, when qdi, qi, and q˙i are known.For a multiple joints system, the position tracking errors must satisfy the following condition in [15]:(16)limt→∞⁡e1=limt→∞⁡e2=⋯=limt→∞⁡en=0.Considering (17), the synchronization errors are defined as(17)ε1=e1-e2+e1,ε2=e2-e3+e2-e1,⋮εi=ei-ei+1+ei-ei-1,⋮εn=en+en-en-1.The expression means that the synchronization errors of the multiple joints system εi are defined as differential position errors among multiple joints system. Setting ε=ε1,ε2,…,εnT∈Rn, function (17) can be written as(18)ε1ε2⋮εn-1εn=2-10⋯0-12-1⋯0⋮⋱⋱⋱⋮0⋯-12-100⋯-12e1e2⋮en-1en=Te,where T is the positive definite matrix.Settings2=ε, in order to achieve HOSM control of sliding variable s2, define σs as(19)σs=s¨2+c1s˙2+c0s2.Substituting (18), there is(20)σs=Ts¨+c1Ts˙+c0Ts.The HOSM control law for sliding variable s2 is designed as(21)u˙vss′=-k3σs-k4σsσs,where k3>0 and k4>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t).The HOSM synchronous control law which is the overall control law is developed as follows:(22)τ′=ueq+M0quvss+M0quvss′.Theorem 2.
Considering system (3), the sliding surface defined in (19) is chosen. When the parameters satisfy k1>0, k2>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t), k4>M˙0-1(q)ρ(t)+M0-1(q)ρ˙(t), and k3>0, the position tracking error e and synchronization error ε can converge to zero simultaneously, under the control effort of the control law given in (22).Proof.
Choose the following Lyapunov function:(23)V=12σTσ+12σsTT-1σs.Differentiating V with respect to time yields(24)V˙=σTσ˙+σsTT-1σ˙s.Considering (14), and differentiating (6) and (20), we get(25)σ˙=u˙vss+u˙vss′+M˙0-1qρt+M0-1qρ˙t,σ˙s=Tu˙vss+Tu˙vss′+TM˙0-1qρt+TM0-1qρ˙t.Substituting (25) into (24), we have(26)σTσ˙=σTM˙0-1qρt+M0-1qρ˙t-k1σ-k2σσ-k3σs-k4σsσs=σTM˙0-1qρt+σTM0-1qρ˙t-σTk1σ-k2σ-σTk3σs-σTk4σsσs≤σTM˙0-1qρt+σTM0-1qρ˙t-σTk1σ-k2σ-σTk3Tσ-k4σTTσσs=σTM˙0-1qρt+M0-1qρ˙t-k2-σTk1σ-σTk3Tσ-k4σTTσσs<-σTk1σ-σTk3Tσ-k4σTTσσs,σsTT-1σ˙s=σsTT-1TM˙0-1qρt+M0-1qρ˙t-k1σ-k2σσ-k3σs-k4σsσs=σsTM˙0-1qρt+σsTM0-1qρ˙t-σTTk1σ-σTTk2σσ-σsTk3σs-k4σs≤σsTM˙0-1qρt+M0-1qρ˙t-k4-σTTk1σ-σTTk2σσ-σsTk3σs<-σTTk1σ-σTTk2σσ-σsTk3σs.From (18), T is positive definite matrix, so it can be obtained that σTσ˙<0 and σsTT-1σ˙s<0. It is proved that the system is stable simultaneously, and the convergence of the position tracking error e and synchronization error ε is assured; that is, limt→∞⁡s=0 and limt→∞⁡ε=0. Figure 7 presents the structure of the HOSM synchronous controller. It can be found that the synchronous control signal is added to the HOSMC control signal.Figure 7
Structure of the HOSM synchronous controller.


## 4. Simulation Results

To demonstrate the performance of the proposed approach in Section3, simulations are performed by MATLAB. In this section, the HOSM controller and HOSM synchronous controller are used to control the dynamic model of the wheelchair, respectively.The values of kinematic and dynamic parameters are assigned asl1=0.3 m, l2=0.3 m, l3=0.405 m, l4=0.3 m, l5=0.18 m, l6=0.12 m, and l7=0.12 m. The radius of wheel is R=0.12, and the radius of small wheel is r=0.05 m. The step parameters are h=0.17 m and d=0.25 m. Moreover, δ=0.66 rad, m1=5 kg, m2=4 kg, m3=3 kg, m4=3 kg, m5=6 kg, m6=3 kg, and m7=3 kg, and j1=0.6 kg·m2, j2=0.7 kg·m2, j3=0.03 kg·m2, j4=0.03 kg·m2, j5=0.04 kg·m2, j6=0.03 kg·m2, and j7=0.03 kg·m2. The rotational speed of climbing mechanisms v=1.675 rad/s. The dynamics of the wheelchair is obtained in Section 2 by using these parameters. Choosing the process of the wheelchair climbing two steps, the desired position trajectories are designed according to analyzing the kinematic model in Section 2.In this section, the position responses and tracking errors of Joint 1 and Joint 2 achieved by the two controllers are shown and compared. For comparison purpose, the simulation is implemented under the same condition by using the HOSM controller and the HOSM synchronous controller, respectively. The parameters of the two controllers arec0=61, c1=70, k1=80, k2=120, k3=100, and k4=120. They are properly chosen to assure that the system can get similar performance using different controllers, and they keep constant during the whole simulation.First is the moment of inertia uncertainties simulation. The inertia of Joint 1 keeps constant during the whole simulation. The inertia of Joint 2 has been changed at 4 s, 8 s, and 12 s. The value of the step amplitude is 0.7 kg·m2; that means the inertia of Joint 2 increases or decreases by 100% every time.Figures8(a) and 8(c) show the position responses achieved by HOSM controller. The results of HOSM synchronous controller are shown in Figures 8(b) and 8(d). It can be seen that the position responses of the two joints are always tracking the desired trajectories effectively by using the two control schemes. There is no obvious overshoot at the startup stage, and the tracking curves of the synchronous controller are smoother at the startup stage.Figure 8
Position responses under inertia uncertainties.
(a)
Position response of Joint 1 (HOSMC)
(b)
Position response of Joint 1 (HOSM synchronous controller)
(c)
Position response of Joint 2 (HOSMC)
(d)
Position response of Joint 2 (HOSM synchronous controller)The tracking errors and synchronization errors are shown in Figure9. When the inertia of Joint 2 changes abruptly, the maximum position error of Joint 1 reaches 2×10-4 rad (Figure 9(a)) using the HOSMC strategy, and the maximum position error of Joint 2 reaches 4×10-4 rad (Figure 9(c)) at the same time. When the HOSM synchronous controller is applied, the maximum position errors of the two joints are 1.4×10-4 rad (Figure 9(b)) and 2.2×10-4 rad (Figure 9(d)), respectively.Figure 9
Simulation results with inertia uncertainties.
(a)
Position error of Joint 1 (HOSMC)
(b)
Position error of Joint 1 (HOSM synchronous controller)
(c)
Position error of Joint 2 (HOSMC)
(d)
Position error of Joint 2 (HOSM synchronous controller)
(e)
Synchronization error (HOSMC)
(f)
Synchronization error (HOSM synchronous controller)The simulation results are listed in Table1. From the comparison of the data above, the system’s tracking errors and synchronous error are reduced over 30% and 50%, respectively, by adopting the HOSM synchronous scheme. It is clear that the proposed scheme has a better control performance with respect to inertia uncertainties.Table 1
Simulation results in case 1.
Item
HOSMC maximum value (rad)
HOSM synchronous controller maximum value (rad)
Position error of Joint 1
2
×
1
0
-
4
1.4
×
1
0
-
4
Position error of Joint 2
4
×
1
0
-
4
2.2
×
1
0
-
4
Synchronization error
8.1
×
1
0
-
4
3.75
×
1
0
-
4Then, the torque disturbance simulation is carried out. The system has torque disturbance and parameter uncertainties in this case. Torque disturbance is a square wave with a period of 10 s, and the amplitude is 50 N·m.The parameters of the two controllers are the same as case 1. Figures10(a)–10(d) show the position responses of the two joints. It can be found in Figure 11 that both controllers give response to the disturbance and uncertainties, but the proposed controller gives smaller error curves. After the disturbance is added, the maximum error of Joint 1 reaches 3.1×10-4 rad (Figure 11(a)) using the HOSMC strategy, and the maximum error of Joint 2 reaches 4×10-4 rad (Figure 11(c)) at the same time. When the HOSM synchronous controller is applied, the maximum errors of the two joints are 2×10-4 rad (Figure 11(b)) and 2.1×10-4 rad (Figure 11(d)), respectively. Figures 11(e) and 11(f) show the synchronization error curves.Figure 10
Position responses under inertia uncertainties and torque disturbance.
(a)
Position response of Joint 1 (HOSMC)
(b)
Position response of Joint 1 (HOSM synchronous controller)
(c)
Position response of Joint 2 (HOSMC)
(d)
Position response of Joint 2 (HOSM synchronous controller)Figure 11
Simulation results with inertia uncertainties and torque disturbance.
(a)
Position error of Joint 1 (HOSMC)
(b)
Position error of Joint 1 (HOSM synchronous controller)
(c)
Position error of Joint 2 (HOSMC)
(d)
Position error of Joint 2 (HOSM synchronous controller)
(e)
Synchronization error (HOSMC)
(f)
Synchronization error (HOSM synchronous controller)The simulation results in case 2 are listed in Table2. From the comparison of the data above, the system’s position errors and synchronous error are reduced over 36% and 40% by adopting the HOSM synchronous scheme. It is clear that the HOSM synchronous controller has stronger robustness and higher performance. It also validates the correctness of the proposed scheme in this paper.Table 2
Simulation results in case 2.
Item
HOSMC maximum value (rad)
HOSM synchronous controller maximum value (rad)
Position error of Joint 1
3.1
×
1
0
-
4
2
×
1
0
-
4
Position error of Joint 2
4
×
1
0
-
4
2.1
×
1
0
-
4
Synchronization error
9.5
×
1
0
-
4
5.5
×
1
0
-
4


## 5. Conclusions

Considering a novel stair-climbing wheelchair with inertia uncertainties and torque disturbances, a HOSM controller is established, and closed-loop stability of the system is proved. Considering the synchronization of the system, a synchronous controller which combines the HOSMC techniques and cross-coupling techniques is proposed, and closed-loop stability of the system is proved, too. The simulation is performed in MATLAB to demonstrate the effectiveness of the proposed controller. The simulation results show that the proposed scheme can give better tracking performance with inertia uncertainties and torque disturbance and better synchronization.



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*Source: 680809-2015-10-21.xml*