A Multi-Joint Single-Actuator Robot: Dynamic and Kinematic Analysis

A Multi-Joint Single-Actuator Robot: Dynamic and Kinematic Analysis A.Nouri, and M.Danesh. Abstract- Changing the length of the arm leads us to a new approach; we name it multi-joint single actuator. To find out the control approach, the mechanism is described in this article. Using one actuator for lots of joints and each time connecting one of them to this actuator is required 2 issues. The first issue is to control the freedom of joint. Another is transferring the torque to the desired joint. In this article, a mechanism is proposed for satisfying both. For the second issue, two approaches are proposed and analyzed. Equations for both of are found. Keywords Robot, mechanism, joint, control, arm. I. INTRODUCTION ECHANICAL arm [1] is a robot act like human arm Mand, is used in different functions like welding, painting, assembling and rehabilitation [2]. In these robots each joint has its own actuator and controller. For controlling several degrees of freedom, same number of actuators is needed like snake robots [3, 4, 5] which have several actuators. Another kind of robots is continuous robot [6, 7] which consists of three actuators. They move a continuous arm like elephant nose. Joints are not controllable separately in continuous robots. Controlling several joints with one actuator is an issue, is investigated in this paper. Two mechanisms is proposed for this purpose. The first one is for selecting the desire joint and other one for transferring the torque to that joint. For transferring the force to the joints there is two ways. The first one is wire and some springs and the second one is using a series of gears. In this article mechanical design and dynamic analysis of both is proposed but gears is used for transferring. This robot can be used in many applications which there are several joints and each time one of them is moving. This mechanism helps us to reduce the number of actuators to 3 for several joints. The only factor that limits us in number of joints is the accuracy of our devises like our manufacturing machineries and our actuator. One of the application of this mechanism is in endoscopy [8] and Laparoscopic [9] since the robot can work in different workspaces. By improving this mechanism in future works we can use it in multi actuator robots like humanoid robots [10]. II. MECHANICAL DESIGN The proposed mechanism consists of two coaxial groups of joints, inner group and outer group. In the inner group, each joint axis has a non zero angle with respect to the others, however, all joints axes of the outer group are in the same plane, i.e., each joint axis has a zero angle with respect to the others. If the inner group is rotated, the difference in its joints angles makes the mechanism to bend merely in the desired joint. For example, for bending the third joint, the inner group must rotate until the angle between the third inner joint axis and the third outer one becomes zero. On the other hand, it can t bend in the other joints (e.g. in the fourth joint) because there is a non zero angle between the other inner joint axes and their corresponding outer ones. For manufacturing this robot, one polyamide cylinder was located inside the other one. Then both of them were cut in the same length. The links were built and joined together, as shown in Fig.1. Then the inner joints were put inside the outer ones. To rotate the desired joint by exerting the actuator force, we investigate two approaches. The first approach is based on using the wire and spring. A wire is along the robot and one of its ends is connected to the end of the robot and its other end is connected to the actuator. When the actuator pulls the wire, its length decreases and makes the robot to rotate. On the other side of the robot (the side with no wire), there are some springs. If the pulling force of the wire is removed then these springs release and the robot rotates to the other side. This is similar to the method used in some continuous robots. Fig.1 angle of inner and outer axis ISBN: 978-1-61804-226-2 96

The second approach is based on using a series of gears which the last gear attached to the robot body and the first gear is connected to the actuator shaft. All of the other gears are free to turn and some of them are located on the joints (one gear on Fig.3. A 3D coordination (1) (2) Fig.2 Cartesian coordination each joint).for example, if the third joint is allowed to move, the gear on this joint acts like a gear attached to the third link. When the actuator s shaft rotates, only the gears before the third joint rotate. This leads to the arm movement in the third joint. Returning the spring to its normal extension is not controllable, thus, the second approach is more accurate than the first approach. On the other hand, in the second approach the ratio of gears can be chosen by designer to control the velocity and the force of each joint. III. DYNAMIC ANALYSIS A. ROBOT WITH WIRE AND SPRING KINEMATIC Consider Cartesian coordination defined in the plane that the robot turns in it, according to Fig.2. In Fig.2, link i is the last movable link and n is the total number of robot links. is the angle between the movable link axis and horizontal axis, x. (x,y) shows the position of the end effecter. is limited by the structure. Due to the links structure in this prototype, the limit is 60 degree. The end effecter position is derived as A 3D coordination is shown in Fig.3. x axes are the same in both recent figures, however, y axes are different in them. The end effecter position in this coordination is The relationship between and other variables can be derived geometrically. Fig.4 shows the geometrical model of this robot. a is the perpendicular distance between the wire and the link axis, and b is the half of the distance between two adjacent joints.c is the hypotenuse of the right-angled triangleabc. F s shows the spring force of the joint i. F c is the wire force supplied by the actuator. is the angle between the two adjacent c hypotenuses in the wire side. β is the same angle, however in the spring side. and β are calculated by the following equations. (3) (4) (5) ISBN: 978-1-61804-226-2 97

Fig.5. Typical characteristics of different kinds of springs Fig.4. different angles in spring and wire (7) The spring and wire torques are (8) Spring force in each group of springs is calculated by the following equations. hard Liner soft (10) n>1 In this robot, x s is calculated by The joint torque is (9) To compute the moment of inertia in each mode of robot, a Catia model for each link is developed. Based on this model, the moment of inertia of each link around its z axis (J n ) and then the moment of inertia of all movable links around the movable joint are calculated as (11) There are three different kinds of springs, hard, soft and linear, and their typical characteristics are shown in Fig.5. M is the mass of movable joints and m is the mass of each link. (12) ISBN: 978-1-61804-226-2 98

From the following equations (16) Then, (17) (18) (19) (20) Fig.6. forces and moments in robot (21) DYNAMICS Dynamic analysis can be done with both Newton-Euler and Lagrangian. Kinematic energy comes from the rotation of arm. (22) NEWTON-EULER: In the following formulation, the torque is calculated. We rewrite all of the formulation with ө. Then (23) (13) At last Q is the generalized force. LAGRANGIAN: (24) In this part, the robot dynamic relations are derived by Lagrangian. From Lagrangian formula and its ingredients, we have (14) In this robot, we have 2 kinds of potential energy, spring and gravity. (25) (15) ISBN: 978-1-61804-226-2 99

description Length Force Force Mass of each link Gravity Angle Links number Moment of inertia symbol L F X F Y m g i j value 0.05 2 1 0.117 9.81 π/2 1 0.00004149 Table 1. Robot characteristics B. ROBOT WITH GEARS KINEMATICS: For gears, it is easier to find the desired arm torque. Because the actuator moment is directly transferred to the joint and can be used directly in the formulation, thus, we rewrite the equations as Robot kinematic with gear is the same as that of with spring and wire. Newton Euler: (26) Here, the potential energy includes only gravity term. (29) (27) (30) (31) Lagrangian: (28) Therefore, the dynamic equation is as DYNAMIC ANALYSIS: (26) IV. THEORITICAL RESULTS We simulate all the equations in MATLAB and get these plots as the result (a)torque(n.m) plot for joint 1 ISBN: 978-1-61804-226-2 100

(b)torque(n.m) plot for joint 2 (d)torque (n.m) plot for joint 4 Fig.6. Theoretical results, (0)=-pi/3, = (0) +t pi/10; V. EXPERIMENTAL RESULTS To find out the effectiveness of the proposed robot design, as it is depicted in fig.7, we made this robot and tested the mechanism. The robot parameters are mentioned in table 1. All of the robots links are made of polyamide which is suitable for production procedure of this (c)torque(n.m) plot for joint 3 Fig.7. the prototype of robot prototype. Polyamides are very hard and need to be cooled in milling process. ISBN: 978-1-61804-226-2 101

In this robot, a RX-64 motor is used for turning the arm in each case. This motor is controlled via MATLAB (software) and is commanded through USB2DYNAMIXEL connector (hardware interface). All of the feedback data are stored in a matrix and are plotted. Fig.8. presents a comparison between the actual torque and theoretical torque for the same trajectory. Although we tried to consider all factors in the theoretical calculation, however, we can t ignore the friction and some production faults effects. In this mechanism, manufacturing accuracy is very important. If inner and outer joints are not exactly coaxial then some difficulties appears in turning the inner joints. Also the axes of the inner joints and outer joints must be in the same plane. In the case, the angle between the axis of the 3 rd inner joint and the axis of the 3 rd outer joint becomes zero, if they do not coincide, they won t move. If the joints have backlash or free space, it causes unwanted turn in other joints. (c)torque (n.m) plot for joint 3 (d)torque (n.m) plot for joint 4 (a)torque (n.m) plot for joint 1 Fig.8. experimental results, (0) = -pi/3, = (0) +t pi/10; V. 3BCONCLUSION In this paper, a novel mechanism is presented to robot design, manufactured and practically tested. This robot has the following contributions. (b)torque (n.m) plot for joint 2 1) Several joints can be controlled with just one actuator and turned with only one actuator; however, in a simple arm an actuator is needed to move each joint. 2) The work space of a continuous arm or a simple robot arm is limited. In this robot, there is a work space for each mode. 3) The mechanism, used in this robot can be used in other robots, especially if there is a need for adjustable joints. 4) Changing the radiuses of gears in this robot can transfer different torques to the joints and it is useful ISBN: 978-1-61804-226-2 102

when we want different torques in each link with the same actuator. In this robot, the accuracy in the production process is very important. In comparison with a robot with the same number of joints, this robot costs less. Acknowledgment This study is partially supported by Isfahan University of Technology. References 1. M. Hiromi, Kinematics for the whole arm of a serial-chain manipulator, Advanced Robotics, Volume 15, Number 2, pp. 255-275(21), 2001. 2. L. Wang, Z. Deng, L. Zhang and, Q. Meng, Dynamic Analysis of Rehabilitative Arm Robot, Mechatronics and Automation, Proceedings of the IEEE International Conference, Luoyang, Henan, 2006, pp. 1181-1185. 3. S. Ma, Li W.J., and, Y. Wang, A Simulator to Analyze Creeping Locomotion of a Snake-like Robot, Robotics and Automation, ICRA. IEEE International Conference, 2001, pp. 3656-3661 vol.4. 4. A. Akbarzadeh and, H. Kalani, Design and Modeling of a Snake Robot Based on Worm-Like Locomotion, Advanced Robotics, Volume 26, issue 5-6, pp. 537-560, 13 Apr 2012. 5. Y. Takita, M. Hasegawa and, M. Nunobiki An investigation of climbing up stairs for an inchworm robot, Advanced Robotics, Volume 15, issue 2, pp. 245-253, 2001. 6. M. W. Hannan, I. D. Walker, Analysis and Initial Experiments for a Novel Elephant s Trunk Robot, IEEE Conf. on Intelligent Robots and Systems, pp. 330-337, Takamatsu, 2000. 7. K. Nakajima,T. Li, H. Sumioka, M. Cianchetti, R. Pfeifer, Information theoretic analysis on a soft robotic arm inspired by the octopus, Robotics and Biomimetics (ROBIO), IEEE International Conference, pp. 110-117,Karon Beach, Phuket, 2011. 8. H. Yamashita, k. Daeyoung, N.Hata, T. Dohi, multi slider linkage mechanism for endoscopic forceps manipulator, Intelligent Robots and Systems, (IROS 2003), IEEE/RSJ International Conference, 2003, pp. 2577 2582 vol.3. 9. B. F. Allen, B. Jordan,W. Pannell and, C. Lewis Y. Takita, M. Hasegawa and, M. Nunobiki Laparoscopic Surgical Robot for Remote In Vivo Training, Advanced Robotics, Volume 24, pp. 1679 1694, 2010. 10. A. Karimi, M. Danesh, A. Tabibian, A. Nouri, Dynamic analysis and path planning for a redundant actuated biped robot, Control, Instrumentation and Automation (ICCIA), 2nd International Conference on, Shiraz, 2011, pp. 1074-1079. ISBN: 978-1-61804-226-2 103