Muhammad Umair Khan

A Novel Spherical Actuator for Robotic Shoulder Joint M. U. Khan1, M. Rizwan2 1- Al-Khawarzimi Institute of Computer Science, UET, Lahore, Pakistan 2- Department of Mechatronics and Control Engineering, UET, Lahore, Pakistan Corresponding Author:- Abstract The usefulness of robotic limbs can be determined by their performance in real life situations. For a robotic limb to affectively replace a human limb, it should at least mimic its natural range of motion and torque. Currently existing systems fall short in satisfying both or at least one of these parameters. To get the robotic dynamics to be as close as possible to the natural dynamics of a human limb, we have designed a novel spherical actuator in place of standard motors and mechanical joints. The designed machine can perform rotation around three separate axis independently and simultaneously, while providing a maximum torque of 1.3Nm. Keywords: Spherical Motor; Magnetic Field Strength; Torque; Degrees-of-Freedom 1. Introduction Although most machines can work faster and bear much more load than humans, scientists have always tried to make their movements as close to biological beings as possible. Humanoid robots are results of such efforts. Humanoid limbs might be able to lift very heavy loads but they have a serious disadvantage when it comes to fluidity and range of motion. A human shoulder is capable of rotation along three separate axis, each with different range of rotation. And in humanoid robotic arms, to move or rotate a joint in multiple directions, more than one actuators are required, each pertaining to a certain specific degree of freedom. This brings with itself, a number of drawbacks. The overall size of the machine is increased due to inclusion of multiple motors. And control of multiple motors, for actuation of a single joint, requires more processing cost. In effect, it is not an optimal method of achieving multiple-DoF. Commercially available motors are linear or rotary with one degree of freedom. But through some recent researches in the past, a new type of actuators has emerged, making rotation possible with multi degrees of freedom. These actuators are often referred to as Spherical Motors having actuation possible with two or three degrees of freedom (DoF) depending upon design. A number of such actuators have been developed by different researchers having different constraints, the details of which are discussed in the following. An alternative design based on the concept of a spherical motor presents some attractive possibilities by combining pitch, roll, and yaw motion in a single joint. In addition to the compact design, the spherical motor results in relatively simple joint kinematics. In some applications, such as high speed plasma and laser cutting, the demands on the workspace and the force/torque requirements are low but the end effector is oriented quickly, continuously and isotropically in all directions. Unfortunately, the popular three-consecutive rotationaljoints possess singularities within its workspace, which is a major problem in trajectory planning and control. A spherical induction motor was conceptualized. Complicated three phase windings must be mounted in recessed grooves in addition to the rolling supports for the rotor in a static configuration. These and other considerations lead to an investigation of an alternative spherical actuator based on the concept of variable reluctance stepper motor which is easier to manufacture [3]. The tradeoff, however, is that a sophisticated control scheme is required. Mashimo et al. constructed an ultrasonic spherical motor capable of 3-DoF [4]. The maneuverability of the design was exceptional but it provided very small amount of torque and thus was unsuitable for high load applications. The mechanical design of a spherical induction motor is complex. Figure 1. Multi-DoF System with Multiple Motors [1] In contrast, a number of other technologies have also been tinkered with to achieve life-like motion in robotic joints. An application of piezoelectric actuator for multi-DoF system was studied by Ying Wu in [5]. Also with recent studies in artificial muscle development, including works by Mirvakili and Hunter in [6] and [7], technology in artificial joints and robotic limbs has advanced by leaps. But still these methods don’t provide as much range of motion as required in most applications. Apart from different constructions of spherical motors, researchers have also worked on improving control and different output parameters of these motors. Ankit Bhatia in [8], [9] used a six-stator spherical induction motor for locomotion and providing dynamic stability to a mobile robot. Hongfeng Li in [10] applied thermal network model to analyze how different speed and load influence temperature rise. They thus improve structure of stator for better heat dissipation. Nagayoshi et al devised a technique that helps decide current distribution to electromagnets for maximum torque. The devised system was also tested and its results confirmed through experimentation [11]. An improved topology was designed and optimized in [12] for maximization of power to volume ratio in a powered-wheel using NSGA-II, which is a multi-constraint/multiobjective evolutionary algorithm. Hongfeng Li et al, in [13], also analyzed the effects caused by different motor structures on end leakage magnetic field. They performed simulations and then verified the results practically on a spherical motor with its permanent magnets in halback array configuration. Bin Li et al formulated a methodology for deriving current excitation strategy and verified the results by finite element analysis and through practical operation [14]. A vision-based feedback was used by applying image processing, namely pattern processing algorithms, and then using a PD controller for stabilizing the system [15]. The applied algorithms were able to identify rotor’s position in space and thus provided an active feedback to the system. Min Dai et al investigated characters of air gap flow-field in a magnetically levitated spherical motor [16]. Dependency of the air gap flow-field was studied on motor’s static and dynamic characteristics. In this research, we focus on removing the discussed drawbacks of existing systems by designing a spherical motor with a novel stator construction. The study includes simulations of magnetic field and output torque based on different coil parameters and subsequently a mathematical model of the optimized system. A PID controller and a unity feedback was also implemented for obtaining closed loop response of the system. 2. Proposed Motor Design The focus of this research is to develop a humanoid robotic shoulder joint. To achieve this goal, we first have to look at the motion constraints and parameters governing the movement of a healthy human shoulder joint. According to Namdari et al. in [17], an average adult can move his/her arm through about 180o in coronal plane, 200o in sagittal plane and 90o about its own axis. Furthermore, for an individual of average height and mass, the shoulder joint should at least produce 1Nm of torque to move an arm. For a humanoid robot to closely mimic an actual arm, the designed actuator must output similar numbers. In light of the previous researches done in this domain and the designs tested henceforth, a specific pole shape was designed to maximize output torque. 2.1 Rotor The rotor is spherical in shape with curved magnetic bars stacked side by side. The magnets are arranged with alternating polarities. This configuration ensures that magnetic flux lines are directed radially from the surface of the rotor. The tightly stacked magnets allow for stranger magnetic field in the surrounding area. Figure 2. Single and Stacked rotor PMs Magnets are stacked side by side to form a sphere like structure. The polarities of the magnets can be arranged in different patterns for different magnetic field distributions. The effect of some arrangements on resultant magnetic field was studied with the help of simulations in EMS for Solidworks. When all the poles are arranged in a unidirectional manner i.e. all magnets have their norths facing outward, the magnetic field plot of the assembly can be represented by the Figure 3. In this case, almost all magnetic field lines are concentrated at the top and bottom edges of the sphere and magnetic field values at points away from edges are almost negligible. Thus, this arrangement will result in a major drawback, as almost no magnetic field lines will pass through the coils, producing no measurable force. 2.2 Stator In the designed system, the solenoids are in the shape of curved rectangles which encircle the PM sphere. The coils can be wound two ways. The direction 1 shown in Figure 5 allows rotation in coronal and sagittal planes, while direction 2 allows rotation in transverse plane. Figure 3. Magnetic Field Intensity plots (a) Unidirectional, (b) Halbach array, (c) Alternating with low pole density, (d) Alternating with higher pole density Figure 4. Stator coil shape On the other hand, if magnets are arranged with alternating polarities, much more magnetic flux lines are passing through the surrounding coils as compared to previous case. Increasing number of poles also increases magnetic field around the rotor. This is due to shorter distance between adjacent pole centers. Another arrangement, that is sometimes used in certain devices, is Halbach array in which polarities are alternated in steps. This leads to a greater spread of magnetic field. It is seen that although magnetic field strength is spread out over a larger area, the average field strength across the coils goes down due to greater gaps between resultant north and south poles. Taking into account the four arrangements of rotor poles, it is established that arranging higher density of magnets with alternating polarities results in highest average magnetic field strength through the surrounding coils and thus, must produce highest amount of torque. Figure 5. Stator coil winding sense 2.3 Complete 3D Model The complete model turns out to look like that in Figure 6. Here, only the coils’ faces that are in front of the rotor are depicted as the rest of the coil has no significant effect on relevant forces. The stator contains a set of 16 coils and the rotor has an equal number of magnets. The coils and their cores are supported on a flower shaped base. Angles α, β and γ represent rotation along x, y and zaxis respectively. As evident from the depicted charged coils, the resultant torque’s normal vector should be along z-axis. This is verified from the results table as the y and x-components of the torque are negligible as compared to its zcomponent. The direction of net torque was verified for different combinations of energized coils. Thus, also verifying that any pose within 3-DoF can be achieved when a specific set of coils is energized to a specific amount. Figure 6. Complete actuator model 3. Inherent Torque of Designed Motor After establishing a proof of concept, and testing different magnet arrangements, a working design was finalized, assembled and simulated. A set of several parameters was calculated including current density through stator coils, magnetic field density inside the structure and force density etc. The design consisted of 16 PMs arranged in a circular pattern with alternating polarities, a set of 16 coils surrounding the PMs hoisted on a flower shaped base and each coil had an iron powder core. Iron powder core provides much lesser losses than soft iron cores and laminated steel cores. This is due to much smaller path allowance for eddy currents. To find maximum holding torque in initial position (no rotation in any plane), four pairs of coils were energized with elements of each pair facing each other. Different parameters like wire diameter, number of turns, packing density and coercivity direction of PMs was set and simulation was run to get results. This process was repeated for different angles of rotation to compare how output torque changed with actuator’s pose. Rotor’s pose is changed i.e. rotation angle about z-axis is gradually increased from 0o to 80o and corresponding maximum torques are calculated as represented in Table 1. To represent how torque is affected by angle of rotation along a single degree of freedom, curve fitting method is implemented based on simulated values and expressions for torque, with respect to rotation, are generated. Expressions (1) and (2), representing second and first order curve fitted polynomial equation respectively, may be used in system’s mathematical model. Table 1. Maximum torque corresponding to rotation about single axis 𝛼 (o) 0 𝜏 (Nm) 1.35 1.24 20 40 60 80 1.05 0.85 0.55 𝜏𝛼 ≅ −5.584 × 10−5 𝛼 2 − 5.63 × 10−3 𝛼 + 1.355 (1) 𝜏𝛼 ≅ −0.0101𝛼 + 1.407 (2) 4. Mathematical Representation We know that force experienced by a current carrying coil, with 𝑛 number of turns, inside a magnetic field is represented as following equation [18]. 𝑓 = 𝑛𝐼(𝑙 × 𝐵) (3) In the proposed design for spherical motor, the carrying current coils are static while the inner magnetic core, housing magnets to generate the magnetic field, is free to rotate. Coils being static, the he rotor will experience a force equal in magnitude but opposite in direction to that on stator coils and will rotate accordingly as it is free to move around. Balancing torques for the system will result in equation (4). 𝛼̈ 𝛼̇ 𝐼𝑎 ̈ ̇ 𝛽 𝛽 𝐽 [ ] + 𝑏 [ ] − 𝑁𝑟𝑛𝑙𝐵(𝛼, 𝛽, 𝛾) [𝐼𝑏 ] = −𝐿𝐹𝑠𝑖𝑛𝜑 𝐼𝑐 𝛾̈ 𝛾̇ (4) With 𝐽 being moment of inertia of rotor, 𝑏 damping coefficient, 𝑁 is number of active coils, 𝑟 is rotor radius, 𝑛 is the number of turns in coil, 𝑙 is affective length of stator coil facing the rotor, 𝐿 is the length of load arm, 𝐹 is total force applied by the load, 𝜑 is incident angle of external force, and 𝛼, β, 𝛾 are yaw, pitch and roll angles respectively. The net magnetic field strength (𝑩) through the coils is dependent on pose of rotor, so it is represented as a function of 𝛼, β and 𝛾. Solving the above equation results in the following relation; 𝛼 𝜃(𝑡) = [𝛽 ] = 𝛾 𝐽(𝑏𝑡⁄𝐽−1+ⅇ 𝑏𝑡 −𝐽 )(𝑁𝑟𝑛𝑙𝐵𝐼𝑥 −𝐹𝐿𝑠𝑖𝑛𝜑) 𝑏2 𝛼 𝜃(𝑡) = [𝛽 ] = 𝐶𝐵𝐼𝑥 + 𝐷 𝛾 𝐼𝑎 𝐼𝑥 = [𝐼𝑏 ] 𝐼𝑐 (5) (6) 𝑠2 (𝑏+𝐽𝑠) (8) The torque (𝜏) induced is represented by the following expression; 𝜏 = 𝑁𝑟𝑛𝑙𝐵(𝐼𝑎 𝛼̂ + 𝐼𝑏 𝛽̂ + 𝐼𝑐 𝛾̂) ( 10 ) 4.1 Closed Loop Response To hold the rotor shaft at a specific orientation, a unity-feedback, closed loop is implemented. Using Matlab, Proportional (𝐾𝑝 ), Integral (𝐾𝑖 ) and Differential (𝐾𝑑 ) coefficients are computed with respect to desired form factor and specific parameters for materials used in the design i.e. neodymium magnets of specific dimensions and copper coils with powdered iron cores. The system is tuned further to acquire a shorter response time and an appropriate transient behavior. After integrating the control and feedback sections in the system equation, its response is plotted for a step input. The overall system transfer function then comes out to be; 𝜃𝑐𝑙𝑜𝑠𝑒𝑑 𝑙𝑜𝑜𝑝 (𝑠) = (𝐾𝑖 +𝑠(𝐾𝑝 +𝐾𝑑 𝑠))(2𝐵𝐼𝑥 𝑙𝑟−𝐹𝐿𝑠𝑖𝑛𝜑) 𝑠3 (𝑏+𝐽𝑠)+2𝐵𝐼𝑙𝑟(𝐾𝑖 +𝑠(𝐾𝑝 +𝐾𝑑 𝑠))−𝐹𝐿(𝐾𝑖 +𝑠(𝐾𝑝 +𝐾𝑑 𝑠))𝑠𝑖𝑛𝜑 ( 11 ) Input Value - PID Controller System Transfer Function Output Pose Figure 7. Control loop block diagram Representing the system in s-domain, we get; 𝑁𝑟𝑛𝑙𝐵𝐼𝑥 −𝐹𝐿𝑠𝑖𝑛𝜑 𝐽𝛼̈ + 𝑏𝛼̇ + 0.0101𝛼 − 1.407 = 0 (7) Here, 𝜃 is the resultant pose of rotor, C is a time dependent variable and D is a constant. 𝜃(𝑠) = for a single degree-of-freedom, the load-free system can also be represented as; (9) For simplicity of calculation and analysis, expression (9) can be replaced by (2) to represent rotation about z axis. Thus, Figure 8 shows the transient closed loop response of controlled system for rotation about a single axis. Although stabilization time is quite reasonable, a slight overshoot occurs which can be eliminated with advanced control techniques. that its x-axis is angled slightly towards the collar in transverse plane as shown in Figure 9. This enables us to cover most of the volume covered by an actual human shoulder joint. Figure 8. System step response 5. Conclusion The simulations verify the conceptual design of proposed spherical actuator. Although, the control feedback was applied for actuation of a single degree of freedom, the system can be easily scaled and the models updated to incorporate rotation in two more planes. Simulations also verify that the designed system can rotate about three axis independently or simultaneously. The angle of rotation for two degrees of freedom is limited if a load arm is attached to the rotor. The third degree of freedom remains freerunning as motion along this axis is not hindered by surrounding coils. Figure 9. Motor fixture on humanoid shoulder This actuator works similar to a ball and socket joint and can be used as such in a humanoid robot. The designed actuator can make a +80o to -80o sweep across its y and zaxis and a complete 360o revolution about its x-axis. To make maximum use of its rotation as a shoulder joint, it has to be placed such In addition to that, a maximum torque of 1.3 Nm is quite small for any real world application, so a spherigear system has to be added at the output stage with a gear ratio respective to the desired torque. This will require some slight changes to shape of rotor shell. In terms of future prospects, optimization may be performed on the shape of EM poles and the thickness of coils for maximum torque to volume ratio. The PM ball may be fixed and the encircling EMs be made movable to study changes in response time and stability of the machine. 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