Proceedings of the IEEE International Conference on Mechatronics & Automation Niagara Falls, Canada • July 2005
A Biologically-Inspired Anthropocentric Shoulder Joint Rehabilitator: Workspace Analysis & Optimization Shabbir Kurbanhusen Mustafa, Song Huat Yeo and Cong Bang Pham
Guilin Yang and Wei Lin Mechatronics Group Singapore Institute of Manufacturing Technology Singapore 638075
[email protected]
School of Mechanical & Aerospace Engineering Nanyang Technological University Singapore 639798
[email protected]
Abstract— This paper presents the design of a biologicallyinspired anthropocentric 7-DOF wearable robotic arm for the purpose of stroke rehabilitation. The proposed arm rehabilitator utilizes the human arm structure to combine with underdeterministic cable-driven parallel mechanisms so as to form completely-deterministic structures. It adopts an anthropocentric design concept, thereby offering the advantages of being lightweight, having high dexterity and conforming to the human anatomical structure. This paper mainly focuses on the workspace analysis of the 3-DOF shoulder module, with respect to the shoulder joint motion range. Workspace parameterization is based on a modified Z-Y-Z Euler Angles approach and utilizing cylindrical coordinates to determine the workspace volume, while workspace evaluation is carried out based on the shoulder joint motion limits and the cable-tension analysis. An effective cable-tension analysis method is also proposed based on the duality between force-closed multi-fingered grasping and cable-driven mechanisms. Finally, the workspace of the mechanism is optimized to match with that of the human shoulder and to obtain a set of dimensions for the shoulder module prototype development.
I. I NTRODUCTION Stroke is a leading cause of severe long-term disability in adults without a routinely available restorative treatment. However, in the last decade, stroke rehabilitation has evolved through an increasing understanding of the neuronal recovery processes. In addition, the introduction of robotic enhancements and virtual reality has further enhanced rehabilitation therapies to carry out intensive and repetitive exercises. In an attempt to enhance the rehabilitation process and supplement the therapists’ work, there has been an increasing interest in the use of robotics for rehabilitation [1-8]. This provides therapists with a means to increase the amount and intensity of the movement of the effected limb, thereby allowing them to focus more on task-specific and complex functional movements. Statistics has shown that most recovery will happen within the first six months after a stroke as this is the period where rehabilitation is most effective due to spontaneous recovery. Thus physical and occupational therapies are concentrated during this period. Beyond this period, outpatient rehabilita-
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tion is given for several more months. Subsequently, little or no rehabilitation therapy is administered to the stroke patients. In addition, when an arm or leg does not recover function during this early period, most patients compensate for the loss by making little or no attempt to use the impaired limb, which ultimately leads to even greater disability. However, recent studies [1-5] have revealed that such repetitive exercises can also be effective, even in the chronic phase, potentially offering hope to millions of stroke survivors. Some of the well-known robotic arm rehabilitators used in rehabilitation research include the MIT-Manus [1] and the ARM Guide [2]. The MIT-Manus is a 2-DOF robot that assists shoulder and elbow movement by moving the hand and the forearm in the horizontal plane. Its unique design feature lies in its low intrinsic end-point impedance that allows the device free movements as well as active-assist therapy. Similarly, the ARM Guide is a singly actuated 3DOF device in assisting reaching movements. Its unique feature is that it is statically counter-balanced so that it does not gravitationally load the arm. However, both have limited capabilities as they have a lower DOF as compared to the 7-DOF human arm. Hence, this limits the motion for rehabilitation in certain planes. A novel idea was also proposed by Kobayashi et al. [6] to develop a muscle suit using McKibben actuators to provide muscular support to the paralyzed. However, it faces numerous challenges in having a limited motion range and issues with slippage and slack of wear. For most of these arm rehabilitators [1,2,7-9], they are designed with a self-deterministic or adjustable selfdeterministic structure which means that the device generates its own characteristic motion that may not coincide with the characteristic motion of the human structure. In addition, most of them are heavy in weight and fixed at a structure or a wall, which limit the motion of the user’s arm. Hence, these devices cannot fully adapt to the user and will eventually place additional stresses on the human joints while carrying out rehabilitation, which may cause greater discomfort and harm to the user. In this paper, we focus on the workspace analysis and
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optimization of the proposed 7-DOF biologically-inspired anthropocentric arm rehabilitator, in particular the 3-DOF shoulder module, based on the shoulder joint physiological limits. The proposed design is inspired by the human arm’s anatomical structure. Its unique feature lies in its synergistic utilization of the human arm structure with non-invasive mechanical structures to form a complete deterministic mechanism that adapts to the human arm and overcomes the noncompliance issue of existing arm rehabilitators. II. H UMAN A RM A NATOMICAL S TRUCTURE The movement of the human arm involves the interaction of the musculoskeletal system, consisting of the bones, linked together by ligaments, through joints whose ends are covered by cartilage and moved by the action of the muscles which are connected to the bones by tendons. A unique property of muscles is that it can only contract but it cannot extend. In addition, muscles have been biologically arranged in a parallel and redundant fashion, as shown in Fig. 1.
Wrist Joint
1-DOF Elbow Module
3-DOF Wrist Module 3-DOF Shoulder Module
Fig. 2. Conceptual design of the Biologically-inspired Anthropocentric Cable-Driven arm rehabilitator [11]
Elbow Joint
Fig. 1.
Obtaining inspiration from the human arm’s anatomical structure, Yang et al. [11] proposed a novel design for a biologically-inspired anthropocentric arm rehabilitator (as shown in Fig. 2). The proposed design combines the human arm bones and joints with kinematically under-deterministic cable-driven mechanisms to form a deterministic biomechanical structure with improved adaptability and wearability. In addition, cables are utilized to have a muscle-equivalent driving scheme. This makes the proposed arm rehabilitator lighter and compact, as all the actuators are mounted on the base, and have greater dexterity due to the presence of redundant cables. From Fig. 2, the 3-DOF shoulder module, the 1-DOF elbow module and the 3-DOF wrist module are connected sequentially to form a parallel-in-series 7-DOF arm rehabilitator.
Shoulder Joint
s scle Mu
s cle us M
III. C ONCEPTUAL D ESIGN
Parallel and redundant muscle arrangement in the human arm
For kinematic analysis, simplifications have been made to model anatomical joints by using revolute, universal and spherical joints, with fixed center of rotations since the movements in the human joints’ instantaneous centre of rotations are negligible [9]. The glenohumeral (GH) joint is modeled as a spherical joint, where the head of the humerus fits into the concave recess of the scapula [10]. The humeroulnar (HU) joint is modeled as a revolute joint whose line of rotation is aligned with the centre of the trochlea, while the radioulnar (RU) joint is modeled as a cylindrical joint whose axis of rotation passes through the distal end of the ulna and the centre of the capitulum [9]. The wrist joint is modeled as a universal joint since the flexion-extension and the radial-ulna flexion axes are believed to intersect. However, the rotational DOF of the RU joint at the elbow has its axes passing through the distal end of the ulna and the centre of the capitulum to the capitate. Thus, the wrist joint is analogous to a 3-DOF spherical joint while the elbow joint is analogous to a 1-DOF revolute joint. Together with the 3-DOF shoulder joint, the basic human arm forms a 7-DOF structure.
The human shoulder joint is designed to generate 3DOF spherical motions. Hence, an under-deterministic cabledriven parallel mechanism is designed such that when combined with the shoulder joint, it will produce a deterministic spherical motion, about the human shoulder joint. In such anthropocentric mechanisms, the critical issue is the matching of the mechanism characteristics with the human body characteristics. For this reason, the first step in achieving such a mechanism is to match the workspace of the mechanism with that of the workspace generated by the human joints. The next section will present the workspace analysis for the 3-DOF shoulder module based on the shoulder joint rangeof-motion limits. IV. W ORKSPACE A NALYSIS The workspace of any mechanism is a pertinent issue in the context of optimum kinematic design. This is especially true in the case of cable-driven parallel mechanisms since the workspace in general is rather limited. The maneuverable workspace is defined as a set of poses where the forward/inverse displacement solutions exist and is within the human arm joint limits without colliding with the human body. In addition, the unilateral property and the flexible nature of cables introduce additional constraints such that
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the cables must always be in tension, which further limits its workspace. For the 3-DOF shoulder module, the critical issues in its workspace analysis are the representation of the human shoulder joint (i.e. GH joint) workspace and the cable tension analysis.
based on the initial pose of the upper arm (as shown in Fig. 4) y’ : Cross product of z’ with x’ z’ : Pointing along the longitudinal axis of the humerus bone at any pose
A. Workspace Representation Discretization algorithms are usually used to numerically analyze the mechanism’s workspace. The procedure consists of discretizing the three dimensional space, solving the inverse kinematics at each pose, and verifying the constraints that limit the workspace. Such discretization algorithms are used by most researchers and can be applied to any type of architecture. In biomechanics, however, workspace representation for the purpose of defining the range of motion is approached differently as compared to mechanism analysis or computer animations. While quaternions seem to be used as the standard method for representing motion in computer animations, they are not so often used in biomechanics. Quaternions have the advantages in terms of avoiding gimbal lock and insensitivity to round-off errors, but they suffer from problems of interpretation in terms of meaningfully clinical or anatomical angles. Euler angles, on the other hand, allow some degree of intuition for the rotation angles, but suffer from singularities. Representations using equivalent angle-axis have also been proposed [12,13], but the main issue for biomechanics is that interpretation of joint motions must have an intuitive feel for clinical studies while being mathematically tractable at the same time. Hence, there is a great debate as to the use of Euler angles, quaternions and equivalent angle-axis for workspace representations [14]. In this paper, we utilize an existing orientation representation approach (known as the modified Z-Y-Z Euler Angles) proposed by Bonev [15], to represent the workspace of the human glenohumeral joint based on its joint range of motion limits. The advantage of this representation is that it is able to separate the 3-DOF orientations into two components: the 2-DOF swing and the 1-DOF twist. The swing component (α and β) determines the direction of the upper arm, while the twist component (γ) determines the rotation about itself, as shown in Fig. 3. This representation is very intuitive for clinical studies and joint motion limit boundary can be easily identified using this representation. The origin of the inertial frame coordinate system is at the glenohumeral joint (as shown in Fig. 4) and the orthogonal axes x, y and z are described as: x : Pointing to the front direction of the body y : Pointing to the upward direction of the body z : Cross product of x with y As for the moving frame coordinate system, its origin is at the glenohumeral joint and the orthogonal axes x’, y’ and z’ are described as: x’ : Pointing along the predefined forearm axis of rotation
Moving Platform in the intial pose
z β
z' γ x'
x
Fig. 3.
α
y
Modified Z-Y-Z Euler Angles description (Adapted from [15])
Longitudinal axis of humerus bone coinciding with z-axis z Forearm axis of rotation is parallel with x-axis
Fig. 4.
y
Reference frame origin coinciding with GH joint origin x
Initial pose of the right upper arm at γ = 0
The orientation workspace of the shoulder module is displayed using a system of cylindrical coordinates (ρ, θ, z), as proposed by Gosselin [16]. For the shoulder module, α, β and γ, will be the polar angle θ, the polar radius ρ, and the z-coordinate respectively. This representation offers an advantage of having no representation singularities when θ = 0 or ρ = 0. Based on the shoulder joint motion range [17], the approximated maximum range of the swing angles and twist angles are α ∈ [0◦ ,360◦ ], β ∈ [0◦ ,140◦ ] and γ ∈ [90◦ ,90◦ ] respectively. These values are approximated as there are slight differences between each individual. 1) Kinematic Modeling: Fig. 5 shows an equivalent kinematic diagram of the 3-DOF shoulder module. The orientation of the frame KM on the moving platform with respect to fixed frame KO , is given by the kinematic transformation TO,M as follows: −−→ R OM (1) TO,M = 0 1 −−→ Where OM = {x y z} is the position of point M with respect to frame KO and R is the rotation matrix based on the modified Z-Y-Z Euler Angles [15] as follows: R Where: ⎡
= Rz (α)Ry (β)Rz (−α)Rz (γ) = Rz (α)Ry (β)Rz (γ − α)
cθ −sθ Rz (θ) = ⎣ sθ cθ 0 0 (Cosine and Sine of respectively.)
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(2)
⎤ ⎡ ⎤ 0 cθ 0 −sθ 0 ⎦ and Ry (θ) = ⎣ 0 1 0 ⎦ 1 sθ 0 cθ the angles are denoted by c and s
M {KM}
boundary for the workspace, which is a cylinder of height π (i.e. -π/2 to π/2) with an irregular shaped cross-section. Fig. 6 shows the irregular shaped cross-section of the right glenohumeral joint workspace and the corresponding region in the Cartesian task space. The other issue is the cable tension analysis at each pose within this workspace boundary. Due to the unilateral properties and flexible nature of the driving cables, it is vital for the cable-driven mechanism to maintain positive cable tension to balance the arbitrary external wrenches. The cable-driven mechanism is very similar to multi-fingered grasping, both of which have a common unilateral property (maintaining tension for cables while it is maintaining compression for grasping). In this section, a computationally effective cabletension analysis approach is proposed, which is equivalent to the force-closure grasping method derived from convex analysis [18,19].
m: No. of cables (i = 1,...,m) Bi: Point i on base Mi: Point i on moving platform Li: ith cable Length Vector bi: Position vector of point Bi wrt point O pi: Position vector of point Mi wrt point O {KO}: Inertial Frame {KM}: Moving Frame
Mi Li
pi
Bi
O bi
Fig. 5.
{KO}
Spherical joint
Equivalent Kinematic Diagram of the 3-DOF Shoulder Module
2) Kinetostatic Equilibrium: In order to withstand any external wrench WExt (i.e. external moments about spherical joint) applied on the moving platform, cables must be able to create tension forces to achieve kinetostatic equilibrium of the moving platform. The kinetostatic equilibrium of the 3-DOF shoulder module is given by: WExt = A.T
(3)
Where: T = {T1 T2 . . . Tm }T ∈ m×1 : Cable tension vector A = [a1 a2 . . . am ] ∈ 3×m : Cable matrix ai = (pi × li ): Cable vector bi −pi L th i li : Unit vector of the i cable = − L = bi −pi i
Region 4
Region 2
Region 2 Region 1
Region 6
Region 3 Region 7
Region 3 Region 4
Region 5
x (degrees)
Region 1 Region 5
Region 6
Fig. 6. Cross-section of the right glenohumeral joint workspace and the corresponding regions in the Cartesian task space at γ = 0
B. Workspace Volume The workspace volume is computed based on a simple discretization algorithm. The range of α,β and γ are discretized into step sizes of Δα, Δβ and Δγ respectively. We then have an elemental cube of width Δα, length Δβ and height Δγ, where the volume of the elementary cube is determined as: vu = Δα × Δβ × Δγ
y (degrees)
(T ≥ 0)
Region 7
(4)
Therefore the volume of the whole workspace is numerically approximated by summation of volumes of all the elementary cubes in the workspace. This method serves as a good approximation of the workspace volume of the shoulder module. C. Workspace Evaluation With the workspace representation method mentioned above, the next step is to evaluate the workspace by determining the feasible poses. The critical constraints of the shoulder modules are the joint motion limits and the cable tension analysis. By limiting the workspace of the shoulder module to the shoulder joint motion limits, we have determined a
Similar to force-closure grasps, a force-closure cable tension means that all cables can always be made positive regardless of the external wrench. In other words, a pose satisfying the force-closure conditions also satisfies the positive cable-tension conditions. The concept of convexity and the concept of hyperplanes in convex analysis with the duality between multi-fingered grasping and cable mechanisms are employed here to establish the force-closure cable-tension condition. Force-closure Tension Condition: The cable tension is force-closure if and only if there does not exist a vector ν ∈ m , ν = 0 , such that for i = 1, . . . , 4, ν T ai = 0 or have the same sign (Where ai = ith column vector of cable matrix A) For a 3-DOF cable-driven parallel mechanism, it only requires four cables to be completely restrained [20]. Hence, for the 3-DOF
shoulder module with m cables, there are a total m! of m C4 = (m−4)!4! different four-cable configurations that can be utilized to fully constrain the mechanism. For each four-cable configuration, the cable tension analysis is carried out by constructing υ, which is by simply taking the cross product of any given set of two independent columns of
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Aj ∈ 3×4 (j = 1, . . . ,m C4 ) resulting in 4 C2 (= 6) possible υs for each Aj (Aj contains the cable vectors of the four cables involved in a particular four-cable configuration). The force-closure statement is then satisfied by checking that the dot products of various υs with the respective remaining columns of Aj do not have the same sign and are not equal to zero. Based on an inverse kinematics approach, this method results in a fast and simple computation to determine the force-closure cable tension for the 3-DOF shoulder module at each pose in the workspace. Note that if the cables can achieve force-closure (i.e. positive cable tension) at a particular pose, it means that the mechanism does not suffer from any singularities at this pose. Hence, singularity analysis is not required. V. W ORKSPACE O PTIMIZATION A. Performance Evaluation Index For the 3-DOF shoulder module, the objective for the design is to achieve a workspace as close as possible to the actual shoulder joint workspace. Hence, the performance evaluation criteria utilizing the workspace volume is given as (Workspace Matching Index, WMI): Feasible Workspace Volume (5) Total Workspace Volume nf easible (0 ≤ WMI ≤ 1) = ntotal Where nf easible is the number of feasible points in the shoulder workspace and ntotal is the total number of points in the shoulder workspace. WMI value of 1 (ideal case) indicates that the 3-DOF shoulder module is able to achieve all the poses in the workspace defined by the shoulder joint motion limits, while WMI of value 0 indicates that the 3-DOF shoulder module is unable to achieve any pose achievable by the human shoulder. WMI =
the Caratheodory Theorem and Steinitz Theorem. Based on these two theorems, the most effective number of driving cables range from n + 1 to 2n. If the number of cables is less than n + 1, the cables cannot achieve force-closure. If the number of cables is greater than 2n, there will be too many redundant cables introduced, which will complicate the cable driving control scheme. Hence, for the 3-DOF shoulder module, the ideal number of cables to maneuver the moving platform is from 4 to 6. After carrying out extensive workspace analysis based on WMI, we realized that the greater the number of cables, the larger the WMI. A larger WMI means that a larger orientation workspace volume (= nf easible × vu ) is achieved. As a result, six cables are chosen to maneuver the 3-DOF shoulder module. 2) Optimal Number of Cable Attachment Points: After determining the optimal number of cables to utilize, the next step in the optimization process is to determine the optimal number of cable attachment points, in both the moving platform and the base of the shoulder module. Based on an assumption of symmetric design, (i.e. with regular polygonal shapes for the cable attachment points on the base and moving platform), there are a total of 6 six-cable configurations to analyze. Through workspace analysis based on WMI, we found that a 3-DOF shoulder module with six driving cables in a 3-3 (i.e. three attachment points on the base and three attachment points on the moving platform) arrangement gives the largest maneuverable workspace, as shown in Fig. 7. Therefore, this particular configuration is selected for the dimension optimization of the 3-DOF shoulder module. r2
d2
Cables
B. Optimized Configuration The optimization criterion is to achieve a workspace volume as close as possible to the actual shoulder joint workspace, using WMI given by (5). The objective of the optimization is to determine the configuration for the shoulder module variables that will result in the greatest WMI value. The design variables include the number of cables, the cable attachment arrangement and the dimensions of the shoulder module. Optimization is carried out methodically by first analyzing the number of cables, followed by the cable attachment arrangement, and lastly by determining the shoulder module dimensions that will result in the largest WMI value. 1) Optimal number of cables: The first step in optimization is to determine the number of cables that will result in the largest WMI. However, before proceeding, two classical theorems in convex analysis are employed [19] to determine the lower and upper limits for the number of cables, namely
Spherical Joint
d1
Base r1
Fig. 7.
Cable arrangement of the 3-3 six-cable shoulder module
3) Optimal Dimensions: After determining the optimal number of cables and the optimal cable attachment arrangement, the final step in the optimization procedure is the dimension optimization of the six-cable, 3-3 shoulder module configuration. The objective of the optimization is to determine the dimension of the shoulder module variables that
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will result in the greatest WMI value. The design variables for the shoulder module are the length of the base and moving platform constraint links and the radius of the base and moving platform, i.e. d1 , d2 , r1 and r2 respectively (as shown in Fig. 7). With an aim of finding a solution with practical realization, the optimization process is subjected to dimensional constraints to ensure that the 3-DOF shoulder module will not be too large or too small for practical applications and are based on the human arm dimensions. A modified complex search optimization method is utilized in this process [21]. After optimization, the WMI improved from 0.270 to 0.696. The optimal values of r1 , r2 , d1 and d2 are 345 mm, 102 mm, 145 mm and 105 mm respectively. VI. C ONCLUSION In this paper, the main focus is on the workspace analysis and workspace optimization of a biologically-inspired anthropocentric 3-DOF shoulder rehabilitator. The 3-DOF shoulder module is the base of a larger 7-DOF arm rehabilitator. Drawing inspirations from biologically-inspired solutions, an anthropocentric design is proposed to combine an underdeterministic cable-driven parallel mechanism with the human glenohumeral joint to form a complete deterministic structure. The objective of the workspace analysis is to determine the workspace based on the glenohumeral joint motion limits, such that the 3-DOF shoulder module can be designed to match the predefined workspace. A workspace representation based on the modified Z-Y-Z Euler Angles is employed due to its advantages of being able to separate the 3-DOF orientation of the glenohumeral joint into two components (i.e. swing and twist component), as well as it ability to intuitively describe the pose of the upper arm based on the glenohumeral joint angles. A computationally effective cable-tension analysis algorithm is also proposed in view of the duality of the cabletension with multi-fingered grasping. Finally, workspace optimization is carried out to determine the optimal configuration for the 3-DOF shoulder module including the number of cables, the number of cable attachment points and its dimensions. In this paper, the workspace optimization algorithm mainly focuses on the workspace matching issue. Other qualitative workspace measures such as global manipulability, stiffness, and joint loadings will be included to guarantee the performance of shoulder module. Our future research will focus on the prototype development of the 3-DOF shoulder module as well as the control system implementation. This is a part of a large research project to ultimately develop a 7-DOF arm rehabilitator for stroke rehabilitation. ACKNOWLEDGMENT The authors would like to thank Prof. I-Ming Chen from the School of Mechanical & Aerospace Engineering, Nanyang Technological University, and gratefully acknowledge the PhD Scholarship awarded to Mustafa Shabbir
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