A novel 2-DOF hydraulic spherical robot joint

Transcription

1 3rd International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'213) Januar 8-9, 213 Kuala Lumpur (Malasia) A novel 2-DOF hdraulic spherical robot joint Lu. Wang, Liang. Wang, houlong. Fang, Yongang Hu Abstract Multiple degree-of-freedom (DOF) motion of transmission sstems generall depends on several traditional hdraulic motors. It is important to derive new structures that enjo the advantages of hdraulic and also can rotate with multi degrees-of-freedom. This paper presents a spherical robot joint driven b hdraulic that can rotate Omni-directionall in a simple mechanism. This stud analsis the inverse and direct kinematics of the robot joint and presents the singularit and workspace. Kewords hdraulic actuation, kinematics analsis, spherical joint. M I. INTODUCTION ULTIPLE degree-of-freedom (DOF) motion of transmission sstems generall depends on multiple traditional electrical or hdraulic motors. However, multiple degree-of-freedom transmission sstems composed of three one-dof motors have some disadvantages, such as large volume, complicated mechanical structure, low precision, bad dnamic performance and so on. On the other hand, human joints like the shoulder joints have at least three degrees of freedom. A spherical motor has the advantage that it can rotate Omni-directionall in a simple mechanism. Up to now, several spherical motors such as ultrasonic spherical motor [1], spherical stepping motor [2], spherical induction motor [3], and spherical motor [4] driven b electro-magnets have been studied. A spherical motor with multi degrees-of-freedom like a human shoulder joint generall has the following advantages: (1) Making the mechanical sstem smaller and lighter. (2) implified control makes higher speed possible (3) Eas realization of the direct drive mechanism (4) Making high precision mechanical sstem possible A practical spherical motor would revolutionize the designs Lu. Wang is with the chool of Automation cience and Electrical Engineering, Beijing Universit of Aeronautics and Astronautics, Beijing, 1191 China (corresponding author to provide phone: ; fax: ; wanglu8858@gmail.com ). Liang. Wang, is with the Mechatronics Department, chool of Automation cience and Electrical Engineering, Beijing Universit of Aeronautics and Astronautics, Beijing, 1191 China ( wangliang@buaa.edu.cn ). houlong. Fang is with the chool of Automation cience and Electrical Engineering, Beijing Universit of Aeronautics and Astronautics, Beijing, 1191 China ( fangsl@gmail.com ). Yongang Hu is with the chool of Automation cience and Electrical Engineering, Beijing Universit of Aeronautics and Astronautics, Beijing, 1191 China. of mechanical sstems. Meanwhile, the output torque of these spherical motors is not enough for driving the robot joints. Considering its self-weight, robots using motor drive can hardl carr extra weight. As hdraulic actuation has advantages of high power-mass ratio, large output force and rapid response, it becomes feasible to make the robot joint with hdraulic actuated. It is important to derive new structures that enjo the advantages of hdraulic and also can rotate with multi degrees-of-freedom. In this paper we presents a spherical robot joint driven b hdraulic that can rotate Omni-directionall in a simple mechanism. This stud analsis the inverse and direct kinematics of the robot joint and presents the singularit and workspace This paper is organized as follows. ection II describes the complete design. Kinematics and singularities are discussed in ection III and IV. We derive the workspace in ection V, and give some simulation results and conclusions in ection VI. II. DEIGN The proposed novel 2-DOF hdraulic spherical robot joint is illustrated in Fig. 1.The joint is mainl composed b the stator(top and bottom parts), fan-shaped leaf blade, sphere rotor, guide rail, complete period motor, fluid distribution hat, and so on. Fig. 1 The structure of 2-DOF hdraulic spherical robot joint The complete period rotation motor install in the rotor, and the motor and the fan-shaped leaf blade connect, shown as Fig. 2. This causes the leaf blade to be possible rotate the rotor in 36 degree. The leaf blade and the rotor are been fixed b top and bottom stator, that forms two airtight working housing. Because the fluid pressure in the two housing is different, the actuation fan-shaped leaf blade will circle its axis swinging, the rotor will also swing together with it. The top of the rotor 273

2 3rd International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'213) Januar 8-9, 213 Kuala Lumpur (Malasia) coordinates with guide rail, limits its rotation, and thus has guaranteed the leaf blade rotating stable and precise. Both the guide rail connects with the measure equipment, which can measure the angle of two guide rails and stator, thus determination rotor s terminal position. The final result is causes the rotor to be possible to rotate two degrees-of-freedom with the stator. rotor guide rail swing leaf blade rotating Fig. 2 The interior structure of the joint The two degree-of-freedom rotation is coupled and complex, and difficult to analze the structure's movement relations between the two input rotation and the rotor s position. It is necessar to stud the direct and inverse kinematics of the robot joint. III. KINEMATIC ANALYI For the better analsis structure's movement relations, we hide the stator, structure simplification chart as shown in Figure 3. Fig. 3 implified scheme of the joint First of all, we define the Coordinate sstem and the variables. In the initial point, the stator, the rotor, the leaf blade coordinate sstem closel coincide, the rotor spindle terminal's two parallel planes normal with X axis superposition, the fan-shaped leaf blade both sides' column surface spool thread with X axis superposition, shown in Figure 4. Fig. 4 Coordinate sstem of the joint Coordinate sstem [ ] stands for tator coordinate sstem (outside ball); Coordinate sstem [ ] stands for otor coordinate sstem (inner ball); V stands for Leaf blade coordinate Coordinate sstem [ ] sstem (leaf blade). k ( α) stands for coordinate rotating an angle of α around the k axis. ( X ) stands for the z component of the X axis of the V z coordinate sstem [ V ] in coordinate sstem [ ] V [ V ] to coordinate sstem [ ]. stands for the transformation from coordinate sstem. As a result of each structure's particularit, mechanic design can have certain influence to the structure movement. Base on the structure shown in ection II, the two-degree-of-freedom hdraulic robot joint has the following two mechanical restraints: (1) tator's restraint: as a result of the stator plane's restraint, leaf blade's X axis is alwas vertical to the stator coordinate sstem's Z axis. We have ( ) XV z. (2) Guide rail's restraint: as a result of guide rail's restraint, the rotor is unable in the Z axis direction to rotate with the guide rail. Hence, the rotor's X axis will alwas superpose in stator's Y axis. We have ( X ). A. Direct kinematics The 2-DOF hdraulic spherical robot joint terminal's movement is a continual process. Two movements, rotating and swinging, are simultaneousl carries on. For ease of analsis, we decompose the movement into two steps; the fan-shaped leaf blade swings first, and then the rotar motor carries on rotation. The fan-shaped leaf blade swings, rotating an angle of γ around the X axis. Then we have the coordinate V 274

3 3rd International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'213) Januar 8-9, 213 Kuala Lumpur (Malasia) transformation matrix 1 X ( γ) cγ sγ sγ cγ where cγ cosγ and sγ sinγ. And then, the rotar motor carries on rotation, rotating an angle of α around the Z axis. Because of the first mechanical restraints, as the leaf blade rotating with the rotor coordinate sstem, the leaf blade would also rotate an angle of θ around the Z axis. tator s Z axis direction unit vector k in the leaf blade coordinate sstem [ V ] coordinate is 1 k X ( γ) cγ sγ sγ (1) 1 sγ cγ 1 cγ The transformation matrix is cθ cγθ s sγθ s 2 2 ( θ) cγθ s sγ cγcθ sγγ c (1 cθ) k (2) 2 2 sγθ s sγγ c (1 cθ) cγ + sγcθ Because of the second mechanical restraint, the guide rail's restraint, the rotor is unable in the Z axis direction to rotate with the stator coordinate sstem. That means the rotor rotates an angel of α around the Z transformation matrix is cα sα Z ( α) sα cα 1 axis, and the coordinate Finall, we have the coordinate transformation matrix from the otor coordinate sstem to the tator coordinate sstem cαθ c + sαγθ c s sαθ c cαγθ c s sγθ s X k Z cαsθ sαcγcθ sαsθ cαcγcθ sγcθ (4) Due to the second mechanical restraint ( X ), we have ( X ) cαθ s sαγθ c c Further, θ arc tan( cγ tan α) (5) We use the Z-Y-X Euler angle to describe the rotor s final posture. The rotor rotatesα, β and γ angles around the Z, Y and X axes, respectivel. (3) cαcβ cαsβs sαcγ cαsβcγ + sαsγ ZY ' ' X' ( α, β, γ) sαcβ sαsβs cαcγ sαsβcγ cαsγ sβ cβsγ cβcγ ubstituting eq. 2 and eq.3 into ' ' '(,, ) ZY X α β γ ields cαcβ cαsβsγ sαcγ cαsβcγ + sαsγ sαcβ sαsβsγ cαcγ sαsβcγ cαsγ sβ cβsγ cβcγ cαθ c + sαγθ c s sαθ c cαγθ c s sγθ s (6) sαsθ cαcγcθ sγcθ Each element of the two matrixes is correspondentl equal. Hence, sαcβ. Then we have α or β π /2 (8) When β π /2, (6) simplifies to cαsγ sαcγ cαcγ + sαsγ sαsγ cαcγ sαcγ cαsγ 1 c c + s c s s c c c s s s s s + c c c s c αθ αγθ αθ αγθ γθ α θ α γ θ γ θ Each element of the two matrixes is correspondentl equal. Hence, α γ π /2 As π π γ, 4 4 (9), this group of solutions cannot be applied. Whenα, (6) simplifies to cβ sβsγ sβcγ cγ sγ sβ cβsγ cβcγ c c + s c s s c c c s s s s s + c c c s c αθ αγθ αθ αγθ γθ α θ α γ θ γ θ (1) Therefore, we have the solution to the direct kinematics in this mechanism. α β Atan 2( sαγ s, cαθ c + sαγθ c s ) γ Atan 2( sγcθ, sαsθ + cαcγcθ) θ Atan 2( cγ, tan α) with. 275

4 3rd International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'213) Januar 8-9, 213 Kuala Lumpur (Malasia) where Atan 2( x, ) represent for arctangent function of double parametric variable. B. Inverse kinematics imilar with the direct kinematics analsis, we use the Z-Y-X Euler angle and the coordinate transformation matrix to get the inverse kinematics solution. According to the (6), (7), (8) and (9), when β π /2, we have γ ± π /2 π π γ, As 4 4, this group of solutions cannot be applied. According to the (6), (7), (8) and (1), whenα, we have the solutions to the inverse kinematics in this mechanism γ arccos( cβcγ) γ arccos( cβcγ) or α Atan 2( sβ, cβsγ) α Atan 2( sβ, cβsγ) As the inverse kinematics shown, each position has two groups of solutions. In the actual structure, the rotor can move to a given position through two was. For example, the rotor can through swinging an angle of γ, then rotating an angle of α arrive at the predetermined position; the rotor also can swing an angle of γ first, then the rotating an angle of π + α arrive at the same position. IV. INGULAITY It can be obtained b the speed recurrence formula of the links that the Jacobian matrix J ( Θ) with respect to the stator coordinates sstem and the speed of the rotor with respect to the rotor coordinate sstem. cαθ c + sαγθ c s sαγ s α α ω sαθ c + cαγθ c s cαγ s θ J ω θ sγθ s cγ 1 γ γ Then we have the Jocobian matrix cαθ c + sαγθ c s sαγ s Jω sαθ c cαγθ c s cαγ s + sγθ s cγ 1 It is known that det [6]. In this sstem, we have c c + s c s s s J ω ields to boundar singularities det( J ) sαθ c + cαγθ c s cαγ s cθγ s ω αθ αγθ αγ sγθ s cγ which ields the singularit conditions as γ In the actual structure, shown as Fig. 5, when the rotor located at the initial point, the terminal of the rotor will not have the displacement if onl rotate the leaf blade (green part). 1 Fig. 5 ingularit of the joint V. WOKPACE The simplest wa to analze the positional workspace is to choose an orientation, and then find the positions that the platform can reach (and the area of the region the cover) - this is called the constant orientation workspace. With the orientation kept static, for example in the smmetric position shown in Fig. 1, it is straightforward to find out if a particular position is within the workspace. We find the reachable points numericall b computing the inverse kinematics of the robot joint. We define a clindrical grid surrounding the structure with the angle β changing from to 36 with 5 increments, the angle γ from to 36 with 5 increments and the radius for 1. The inverse kinematics solution is then computed for each grid point. If the inverse kinematics solution exists, the point belongs to the workspace of the robot. The envelope of the workspace is then extracted from the collection of reachable points. Fig. 6 Workspace of the joint The spherical robot joint s constant orientation workspace is given in Fig. 6. It can be seen from Fig. 6 that the spherical robot joint can reach ever point on surf of intersection part of circular conical surface and spherical surface. The result of the 276

5 3rd International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'213) Januar 8-9, 213 Kuala Lumpur (Malasia) workspace analsis confirms the feasibilit of 2-DOF hdraulic spherical robot joint. VI. CONCLUION In this paper we presents a two degree-of-freedom spherical robot joint driven b hdraulic that can rotate Omni-directionall in a simple mechanism. For the better analsis structure's movement relations, we have studied the direct and inverse kinematics. The complete singularit and workspace analsis were provided in this work. All this result of analsis confirms the feasibilit of 2-DOF hdraulic spherical robot joint. In order to improve the whole performance of the robot joint, the further dnamic analsis and optimization of parameters is necessar. All the studies in this paper can provide reference for further research and improvement. EFEENCE [1] H. Kanazawa, T. Tsukimoto, T. Maeno anf A. Miake, Tribolog of ultrasonic motor, Journal of Japanese ociet of Tribologists (in Japanese), 38 (3) (1993) [2] T. Yano and T. uzuki, Basic characteristics of the small spherical stepping motor, Proc. of 22 IEEE/J International Conference on Intelligent obots and stems (IO 2), (22) I.. Jacobs and C. P. Bean, Fine particles, thin films and exchange anisotrop, in Magnetism, vol. III, G. T. ado and H. uhl, Eds. New York: Academic, 1963, pp [3] A. Tanaka, M. Watada,. Torii and D. Ebihara, Proposal and design of multi-degree of freedom actuator, Proc. of 11th MAGDA Conference (in Japanese), (22) [4] G.. Chirikjian and D. tein, Kinematic design and commutation of a spherical stepper motor, IEEE/AME Transactions on Mechatronics, 4 (4) (1999) [5] Tomoaki Yano, imulation results of a hexahedron-octahedron based spherical stepping motor, Journal of Mechanical cience and Technolog 24 (21) 33~36 [6] C. Gosselin and J. Angeles, ingularit analsis of closed-loop kinematic chains, IEEE Transactions on obotics and Automation, vol. 6, no. 3, pp ,