Dynamic Simulation of an Improved Passive Haptic Display

Transcription

1 Dynamic Simulation of an Improved Passive Haptic Display A Thesis Presented to The Academic Faculty by Davin Karl Swanson In Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical Engineering Georgia Institute of Technology May 999

2 Dynamic Simulation of an Improved Passive Haptic Display Approved: Dr. Wayne Book, Chairman Dr. Aldo Ferri Dr. William Singhose Date Approved

3 Acknowledgements I would like to first thank Dr. Wayne Book for his guidance and patience through the challenging process of bilingual advisement. Thanks to all of my colleagues that I have bounced ideas off of in the past two years L. J. Tognetti, Saghir Munir, Klaus Obergfell, and Cody Watson in particular. This work was completed with partial support from the National Science Foundation, Grant IIS Special thanks to Eric Romagna for keeping an eye on me during my stint at EN- SAM, and for providing some very entertaining times after hours. Tu seras toujours un vrai copain. iii

4 Contents Acknowledgements iii List of Tables viii List of Figures ix Summary xii Introduction. The Haptic Display Passive Haptic Displays Organization of This Work Background 4 2. Development of PTER PTER - Physical Description PTER - Kinematic and Dynamic Equations Clutch Dynamics Control Methods Dynamic Simulation SimPTER - Design and Development 2 3. Motivation for an Enhanced Simulation iv

5 3.2 SIMULINK Implementation Input Tip Force Controller Clutch Model Clutch Dynamics Friction Model Selection Friction Model Adaptation Clutch Model Implementation Dynamic Simulation Clutch Model Validation Required Changes to SimPTER PTER - Actuator Identification and Modeling 3 4. Motivation for System Identification Previous Testing Initial In-Place Testing Development of a Motorized Clutch Testbed Testbed Setup and Calibration Servomotor/Servocontroller/Tachometer Torque Sensor Clutch Testbed Experiments Clutch Dynamics Friction Properties Dynamic Friction Breakaway Torque v

6 4.7 Comparison - Torque Bar Tests and Motorized Testbed Tests Controller Evaluation 6 5. SimPTER Modifications Simulation Framework Definition of Simulation Runs Measures of System Performance Baseline Controller Controller Definition Simulated Results Proportional Torque Control Controller Definition Simulated Results Physical Implementation Velocity Controller Controller Definition Simulated Results Conclusion and Future Work Conclusion Findings and Contributions Dynamic Simulation Passive Interface Controllers Proportional Torque Feedback Control Velocity Controller Future Work vi

7 6.3. Simulation PTER Hardware and Software A Friction Model Calculation of Clutch Torques 88 A. Case A.2 Case A.3 Case A.4 Case A.5 Case A.6 Case A.7 Case A.8 Case B Friction Model Validation - Reduced-Order Tests 93 B. Clutch B.2 Clutch B.3 Clutch B.4 Clutch vii

8 List of Tables The 2 Possible Modes of PTER (N/A = not applied) Case 7 Subcases Individual Quadratic Fits and Error Parameters Scaled Quadratic Fit and Error Parameters Simulation Results of Baseline and Torque Feedback Controllers Simulation Results of Several Controllers The 2 Possible Modes of PTER (N/A = not applied) viii

9 List of Figures PTER with clutch numbers and link letters PTER - dimensions and coordinate systems SimPTER2 - SIMULINK Model Stick-Slip Friction Model Simple One-Mass Friction System Karnopp Friction Model - Block Diagram SimPTER Dynamic Simulation Block Reduced Order Test - Clutch Reduced Order Test - Clutch with Clutch 2 Locked Torque Bar Test - Experimental Setup Torque Bar Test - Dynamic Torque Data and Curve Fit Torque Bar Test - Breakaway Torque Data and Curve Fit Clutch System ID Testbed - Mechanical Layout Torque Sensor Calibration - Calibration Data and Linear Fit Power Supply Step Response - 3 volt Command Input Power Supply Step Response - 8 volt Command Input Clutch System Step Response - 3 volt Command Input Clutch System Step Response - 8 volt Command Input Simplified Clutch Mechanical Diagram Clutch System Step Response - 2-to-8 volt Command Input Input/Output Response of Data Acquisition System ix

10 22 Clutch Step Response Repeatability - to 6 volt Step Input Clutch Step Response Repeatability - 2 to 6 volt Step Input Typical Dynamic Clutch Torque Test Measured Dynamic Clutch Torque Measured Dynamic Clutch Torque with Individual Quadratic Fits Measured Dynamic Clutch Torque with Single Scaled Quadratic Fit Final Dynamic Clutch Torque Model Typical Breakaway Torque Test Data Typical Breakaway Torque Test Data - Detail Breakaway Torques Measured Breakaway Torques with Quadratic Fit Dynamic and Breakaway Torque Model Comparison Model Comparison of the Two Clutch Tests Tracking Error Definition Division of PTER s Controller SimPTER Output - Tip Position Plot - Baseline Test SimPTER Output - Clutch Velocities - Baseline Test SimPTER Output - Linear Tip Jerk and Clutch Velocities - Baseline Test SimPTER Output - Linear Tip Jerk and Acceleration - Baseline Test 67 4 Clutch Controller with Proportional Torque Feedback Clutch Controller with Proportional Torque Feedback - SimPTER Implementation Proportional Torque Feedback Controller - Position Tracking Performance x

11 44 Proportional Torque Feedback Controller - Smoothness Measure Endpoint Position - Baseline and Proportional Torque Feedback Controller Clutch 2 Torque Following - Baseline Controller Clutch 2 Torque Following - P Feedback Controller (Gain=.6) Implemented Torque Controller - With and Without P Feedback Implemented Proportional Feedback Controller (gain=.2) - Unstable Behavior Simulation of Testbed Torque Feedback Controller Desired Velocity Determination - Implemented Gomes Velocity Controller Desired Velocity Determination - Proposed Velocity Controller Instantaneous Generated Clutch Force Vectors Impedance and Velocity Controller Comparison - Endpoint Position Impedance and Velocity Controllers - Linear Tip Acceleration Magnitude Reduced Order Test - Clutch Reduced Order Test - Clutch with Clutch 2 Locked Reduced Order Test - Clutch Reduced Order Test - Clutch 2 with Clutch Locked Reduced Order Test - Clutch Reduced Order Test - Clutch 3 with Clutch 4 Locked Reduced Order Test - Clutch Reduced Order Test - Clutch 4 with Clutch 3 Locked xi

12 Summary This work studies the performance and control of a passive haptic display. The device is passive in the sense that all its actuators are dissipative they may only remove energy from the system. All energy entering the system must be supplied by a human operator. The purpose of the device is to exert forces on this operator. A dynamic simulation of the device is enhanced through the addition of an actuator model incorporating dynamic response and friction behavior. Experimental data on actuator performance is gathered and used to improve the accuracy of the new actuator model. The simulation is then used to evaluate the performance of two new control concepts. One of these concepts is implemented on a testbed, and experimental results are presented. xii

13 Chapter Introduction. The Haptic Display The word haptic is one unfamiliar to most. It comes from the Greek haptesthai, meaning to touch, defined as relating to or based on the sense of touch. A haptic display is a device that interacts with a user through his or her sense of touch. Just as a computer monitor is a visual display and a set of headphones is an aural display, a haptic device is a touch display. There are many uses for such displays in the fields of teleoperation, artificial environments (virtual reality), and ergonomics. One of the first applications of haptic displays was in teleoperation. Tactile feedback can improve the performance of a local operator manipulating a remote system. Such a system relies on the user projecting his dexterity into the remote environment. Although humans rely highly on visual cues to perform tasks, we also depend on tactile cues for object identification and manipulation. [] In situations where visual sensing is impaired it can actually be replaced by tactile sensation. [8] Also, a more accurate representation of the remote environment will instill a greater sense of presence in the user, which may improve his or her performance. [2] Virtual reality is another application of haptic displays. Force feedback can deliver a third sense along with sight and sound to the virtual experience. There are a wealth of applications in this field, including prototype visualization for CAD, virtual

14 evaluation and testing of factory layout and process flow, haptic input devices, and of course entertainment. In fact it is entertainment that has started to deliver the haptic display into the mainstream, evidenced by the latest force-feedback joysticks now available for home computers. In addition to their force reflection capabilities, haptic displays can also be used for path guidance. In this application, the device tries to guide the user along a specific path and/or constrain movement within a certain region. One such application would be to aid the physically disabled, for example to facilitate walking in a paraplegic by controlling limb movement. [9].2 Passive Haptic Displays Most existing haptic displays are active, comprised of actuators that can do positive or negative work on the interfaced system. However, some current research involves the study of passive haptics. [7] [4] [5] [8] A passive haptic display has no actuators that can add energy to the system. That is, all energy added to such a device must come from the user. Such a device has a primary advantage of safety over a similar active device. Uncommanded movement is less probable in passive devices and is in general easier to prevent. This makes passive haptics ideal for applications where safety has high priority, such as assisted surgery and situations where high contact forces are possible. Passive haptic displays are a challenge to control, since arbitrary control actions are not possible. A control action which adds energy to the system is not achievable. An effective controller must determine whether or not desired control actions are unachievable, and if so, define an achievable set of command inputs which act as a 2

15 compromise between system performance and realizability..3 Organization of This Work This work explores the enhancement of a simulation of a passive haptic display and its subsequent use in the evaluation of control concepts that may increase system performance. This chapter has provided some introductory information on haptic displays. Chapter 2 reviews previous work done in the design and manufacture of a two degreeof-freedom passive haptic testbed, the Passive Trajectory Enhancing Robot (PTER). Chapter 3 explains the enhancement of a dynamic simulation of PTER through modeling of stick-slip friction and actuator dynamics. Chapter 4 deals with the design and manufacture of an actuator testbed used to perform system identification tests on one of PTER s four actuators, with the intent of building an actuator model for use in the dynamic simulation. Chapter 5 addresses the implementation and simulated performance of two new control concepts. Experimental results for one of these control concepts are also presented. Finally, Chapter 6 contains closing comments about the contribution of this work and some ideas for further research. 3

16 Chapter 2 Background 2. Development of PTER 2.. PTER - Physical Description An experimental passive haptic testbed has been built and used by previous students for the purpose of studying the behavior of such a device and to evaluate control techniques. [6] [] This device has been named PTER the Passive Trajectory Enhancing Robot. Figure is a diagram of PTER. It is a planar robotic arm in a five-bar parallel linkage arrangement. PTER s purpose is to exert forces on the hand of a user, who grips the handle at the endpoint of link D. It does this by providing torques to links A and B through its network of four actuators. The actuators are controllable friction clutches. The clutches are passive devices, only serving to remove energy from the system, hence the passivity of the entire device. Clutch and clutch 2 connect links A and B, respectively, to ground. Clutch 3 couples the velocities of links A and B together. Clutch 4 inversely couples links A and B together through the gears located in the middle of the device. Since PTER has more clutches than degrees of freedom, the robot is overactuated. This configuration was selected in order to provide greater freedom in providing arbitrary torques to each of the main links A and B. 4

17 B 3 2 C A 4 D Figure : PTER with clutch numbers and link letters 5

18 In addition to its actuators, PTER contains several sensors. The handle mount holds two strain gauges, used to measure the magnitude and direction of the user applied force. Links A and B are each connected to potentiometers, which are used to determine their positions. Since PTER has two degrees of freedom, the angles of links A and B are sufficient to fully describe the state of the system PTER - Kinematic and Dynamic Equations Kinematic and dynamic analysis of PTER have been sufficiently addressed by Charles [4], Davis [6], and Gomes []. The pertinent equations will be summarized here and the reader is referred to their work for the full derivations. These equations were used in the development of the dynamic simulation of PTER later in this work. Figure 2 is a schematic diagram of PTER, showing the applicable coordinate systems and parameter terminology. Forward kinematics, transforming angular link position to global cartesian tip position: x tip = l A cos(θ ) l D sin(θ 2 ) () y tip = l A sin(θ ) + l D cos(θ 2 ) (2) Inverse kinematics, transforming global cartesian tip position to angular link position: ( x 2 + y 2 la 2 l 2 ) D q = arccos (3) 2l A l D ( ) [la + l D cos(q )]y [l D sin(q )]x q 2 = arcsin (4) x 2 + y 2 θ = q 2 (5) θ 2 = q + q 2 + π 2 (6) 6

19 Endpoint (x t, y t ) l d D r d l b A y r a θ C l c =la x r c r b θ 2 B Figure 2: PTER - dimensions and coordinate systems 7

20 Positions of coupling clutches 3 and 4: θ 3 = θ θ 2 (7) θ 4 = θ + θ 2 (8) It is a straightforward task to differentiate equations 8 with respect to time in order to calculate the angular velocities and accelerations of the coupling clutches. Velocity Jacobians relating link angular velocity and endpoint linear velocity: ẋ ẏ ẋ ẏ = = 2 l A sin(θ ) l D cos(θ 2 ) θ l A cos(θ ) l D sin(θ 2 ) θ 2 l A sin(θ ) + l D cos(θ 2 ) l A sin(θ ) l D cos(θ 2 ) (9) l A cos(θ ) + l D sin(θ 2 ) l A cos(θ ) l D sin(θ 2 ) θ 3 θ 4 () Force Jacobian relating endpoint forces to clutch torques: τ τ 2 = 2 τ 3 τ 4 2l A sin(θ ) 2l A cos(θ ) 2l D cos(θ 2 ) 2l D sin(θ 2 ) l A sin(θ ) + l D cos(θ 2 ) l A cos(θ ) + l D sin(θ 2 ) l A sin(θ ) l D cos(θ 2 ) l A cos(θ ) l D sin(θ 2 ) f x fy () The Jacobian matrices in Equations may be inverted to obtain link velocities from endpoint velocities. Since the Jacobian in Equation is non-invertible, clutch torques cannot be computed uniquely from a given endpoint force. This is due directly to the fact that the system is overactuated. The rigid-body equations of motion for the system, where r w is the distance from the end of link w to its center of mass, m x is the mass of link x, I y is the mass moment of inertia of link y, and τ z is the net torque on link z, are: α = 2 ( ma r A 2 + I A + I C + m C r C 2 + m D l A 2 ) (2) 8

21 β = 2 ( mb r B 2 + I B + I D + m C l B 2 + m D (r D l B ) 2) (3) γ = m C r C l B m D l A (r D l B ) (4) 2α γ sin(θ 2 θ ) θ γ sin(θ 2 θ ) 2β θ Clutch Dynamics + γ θ 2 2 cos(θ 2 θ ) γ θ 2 cos(θ 2 θ ) = τ a τ b (5) The dynamics of PTER s clutches have been briefly examined by Gomes []. His experimental setup was less than ideal, and a satisfactory model was not obtained. It was made clear, however, that the clutches act similar to a first-order system with time delay, and that the lag and time constant of the system, though uncertain, are sufficiently large to affect the control of the robot. 2.2 Control Methods Since its inception, several controllers have been implemented on PTER. PTER s control needs can be separated into three parts. One controller must identify a set of desired link torques given the instantaneous state of the system and its desired behavior. The second controller must transform the desired link torques into a set of achievable clutch torques, taking into account the system state and the clutches passivity constraint. Finally, the third controller must provide command signals to the physical hardware in order to produce the torques desired by the second controller. Most of the previous work in controlling PTER has concentrated on the first two concepts. The latter task of generating control signals has been solved open-loop through the use of a lookup table. The initial controller as suggested by Charles [4] and implemented by Davis [6] is 9

22 an impedance controller, which computes desired tip forces through the simulation of spring and damper elements between the endpoint of the robot and the desired path. An algorithm named the torque translator was developed by Davis in order to select a set of achievable clutch torques which match the desired link torques as closely as possible. The signs of the clutch velocities are used to determine whether or not a desired output torque is achievable. Gomes looked into several different controllers with the aim of both improving path following performance and minimizing tip acceleration []. He implemented a simplified version of the torque translator, which considered desired tip forces rather than desired clutch torques. High levels of tip acceleration due to rapid application and release of clutches was evident. In light of this, Gomes implemented a blending algorithm, which gradually applied and released clutches. This algorithm did not significantly improve the tip acceleration profile. A controller using the tip velocity rather than the position error was also investigated with promising results, though the implementation was very basic. 2.3 Dynamic Simulation A dynamic simulation of PTER was written in MATLAB by Charles [4]. In the simulation the controller attempts to constrain the endpoint of the device to a circular path. A force along the circle and a periodic normal disturbance force are applied to the endpoint. The simulation computes net torque on links A and B by combining required clutch torques with the user input force translated into link torques through the Jacobian. The equations of motion are inverted and used to compute the link accelerations from the net torques. Once the link accelerations are calculated, they

23 are integrated to obtain link velocity and position. The Davis simulation was very rudimentary, having virtually no clutch modeling. When the controller requested an arbitrary torque from a clutch, the simulation assumed that the clutch immediately delivered exactly that torque. The only clutch modeling lied in two checks: The requested torque must not exceed the maximum capacity of the clutch. The direction of the requested torque must not violate the passivity constraint (i.e., must not add energy to the system.) This simulation did a good job at basic validation of controller concepts, but performed poorly at predicting the true behavior of the device, which is influenced by factors not modeled in the equations of motion, such as friction effects and clutch dynamics.

24 Chapter 3 SimPTER - Design and Development 3. Motivation for an Enhanced Simulation Numerical simulation is a powerful tool for enhancing the design process of a multitude of engineering systems. It allows one to evaluate different system configurations and control concepts without having to physically construct the system. Often this yields a cheaper, quicker, more straightforward, and safer development process. It appears that PTER s performance is limited by inherent nonlinearities in its hardware specifically stiction and stick-slip effects at the clutches friction interfaces and in the finite gap between the clutch plates when the clutch is not applied. The latter results in time lag and high initial-contact forces. Both of these nonlinearities create a jerky feel to the user. In order to facilitate the study of control systems and possible alternative actuators, it was decided to enhance the dynamic simulation by implementing a more accurate model of PTER. This chapter provides an overview of the extensive changes made to the original simulation, called SimPTER. The bulk of this work was done at ENSAM Paris, France, with the help of Eric Romagna under the tutelage of Professor André Barraco. [7] 2

25 3.2 SIMULINK Implementation In the end, the new SimPTER was an almost complete rewrite of the original. There is very little in common between the two simulations. The only parts that carried over to the new simulation were the control code and several functions describing the dynamic properties of PTER. These functions do things such as Jacobian transformations and inverse dynamics computation. From this point on, the name SimPTER will apply to the new simulation. The previous simulation was essentially a group of MATLAB M-files which were run from the command line. Since the simulation would eventually be used to evaluate different system configurations, it was decided to implement SimPTER in SIMULINK in order to take advantage of its GUI and improved usability. Figure 3 is a diagram of the SimPTER SIMULINK model. The model is composed of four main blocks: Input Tip Force This block models the input force. Controller This block contains the controller. Clutch Model This block comprises the entire clutch model, including both dynamic response and nonlinear friction properties. Dynamic Simulation This block contains all dynamic information about PTER and computes its full state based on the net torque applied to each link. The remainder of this section more thoroughly explains the purpose and function of these blocks. 3

26 Position theta Applied Torques Demux Net Torque Position 2 Vel Vel 2 Net Torque 2 Pre Vel Pre Vel 2 Dynamic Simulation Clock theta2 x force time y force Input Tip Force time Simulation Time Time Display Mux Input Force(2)/ Position(2)/ Velocity(2)/ Time Current Tau Last 4 Tau 4 Current Tau Last 3 Tau 3 Current Tau Last 2 Tau 2 Current Tau Last Tau Torque Memory app_torque Applied Torques Pulse Generator Applied Torques command_voltages Commanded Voltages Input Vector Clutch Actual Torque Voltages (4) Clutch 2 Actual Torque Clutch 3 Actual Desired Torque Torques Clutch 4 Actual Torque Controller Mux Generated Brake Torques Net Torque on Arm Net Torque on Arm 2 taue Torque Vector tau_in tau_in2 Control Voltages theta theta2 Pre Vel Generated Torques/ status Demux Pre Vel 2 tau4_prev tau3_prev Brake tau2_prev Torques(4)/ tau_prev Model Errors Clutch Model STOP Stop Simulation Figure 3: SimPTER2 - SIMULINK Model 4 tau_d Desired Torques model_error Error in Brake Model model_case Model Cases

27 3.2. Input Tip Force This block provides the simulation with the tip forces and resultant link torques generated by the virtual user on PTER s handgrip. Currently, the force is modeled as having a constant component parallel to the desired path and a tangent component comprised of a summation of sinewaves. This input was chosen to provide a disturbance force to the tip of the robot, without singling out a single frequency which could lead to instabilities or lockup conditions within the controller Controller This block is comprised of all three components mentioned in the previous chapter. That is, one component computes desired link torques, the second transforms the desired link torques into achievable clutch torques, and the third provides command signals to the physical hardware (which is, in this case, the clutch model.) The controller initially implemented in the simulation is the impedance controller and torque translator mentioned in the previous chapter, combined with a simple piecewise linear torque-voltage model to provide open-loop torque control. This look-up table method to compute command voltages was used because it is the same method used in the physical setup Clutch Model This block contains the model of PTER s clutches. The two parameters of the clutches that were to be modeled were the dynamics and the friction behavior. 5

28 Clutch Dynamics Data from both Gomes [] and the clutch manufacturer s data sheet indicate that the clutches act like a first-order system. Due to this insight, a first-order model was selected to represent the clutch dynamics. This is modeled in the simulation by applying a first order transfer function of the form G(s) = t c s + to the control signals provided by the controller block, where t c (6) is the clutch time constant. It was decided to apply the dynamics to the control signal through physical insight into the system. Since the clutch is essentially an electromagnet that provides an attractive magnetic force between its two plates, the input current is directly proportional to the normal force between the clutch plates. Therefore, in the simulation the control signal is proportional to the clutch plate normal force. By applying dynamics to the control signal, it can be thought of as affecting the clutch plate normal force. This normal force is one of the factors which ultimately governs the torques provided by each clutch. Gomes found a large discrepancy in published and measured values for the time constant of the clutches []. The manufacturer reports a time constant of.24 seconds, while Gomes s experiments place the time constant somewhere in the wide range of seconds. Clearly, a more accurate value for the clutch time constant is needed Friction Model Selection Friction plays a crucial role in the operation of PTER, as in it lies the mode of actuation. Because of this, the simulation requires a friction model in order to accurately 6

29 F f F B V F B Figure 4: Stick-Slip Friction Model V F in M F f ///////////////// Figure 5: Simple One-Mass Friction System represent the behavior of the system. A literature search was performed to identify possible candidate numerical friction models. Several models were evaluated, including the Dahl [5], reset-integrator, and bristle models [2]. In the end, the Karnopp friction model [3] was selected due to its clean separation of static and dynamic behavior, ability to model important nonlinearities such as high breakaway forces, and acceptable computation requirements. The reset-integrator model was also seriously considered, but was found more difficult to implement within the existing simulation framework than the Karnopp model. Stick-slip friction is a discontinuous phenomenon (see Figure 4.) It consists, however, of two separate modes, each of which is piecewise continuous for a specific system 7

30 F + - m Vr D v D v x F f D v D v D v D v + + F slip F h F h F stick Figure 6: Karnopp Friction Model - Block Diagram variable. The applicable mode at any point in time depends on whether or not there is relative velocity between the two friction surfaces. The Karnopp model utilizes different sets of governing equations for each of these modes. In effect, the order of the dynamic system is reduced when the relative velocity is equal to zero. Within this reduced-order framework, the position constraint of zero relative velocity is incorporated into the system of differential equations. This provides a practical means of dealing with the friction force discontinuity at zero relative velocity. For a simple single-mass system such as that in Figure 5, the Karnopp model yields the system shown in Figure 6 and the following equations for frictional force F f : F f = g(v ) : V δv F in : V < δv, F in F B F B : V < δv, F in > F B (7) 8

31 When the magnitude of the relative velocity V between the two surfaces is greater than or equal to a very small value δv, the system is said to be in the slip mode. Even though the system is physically slipping when V is not equal to zero, the nonzero region around V = is defined in order to account for close-to-zero errors. In the slip mode the friction force is dependent only on the relative velocity, and is determined by an arbitrary function g(v ). When the magnitude of the relative velocity V between the two surfaces is less than δv, the system is said to be in the stick mode. In the stick mode the system is static, and the friction force F f exactly cancels the driving force F in, unless F in exceeds the breakaway force F B. In the latter case the F f is equal to F B and the body will experience nonzero acceleration. After a short interval the magnitude of V will exceed δv and the model will transition from the stick mode to the slip mode Friction Model Adaptation The basic Karnopp model simulates the friction force between a single moving mass and a surface. In order to apply the Karnopp friction model to PTER s friction surfaces, several modifications were necessary. First of all, the above equations were changed from linear to angular coordinates. Also, a variable normal force dependent on the clutch input current was implemented. Since PTER has four friction surfaces, four separate instances of the model are used in the simulation. Due to these modifications, Equation 7 as used by SimPTER is τ f,x = g x (ω, i) : ω δω τ in,x : ω < δω, τ in,x τ B,x (i) τ B,x (i) : ω < δω, τ in,x > τ B,x (i) (8) 9

32 where x =... 4 represent the four clutches, and i is the input current to each clutch. Notice that all forces from the initial model are replaced with torques, and linear velocity has been replaced with angular velocity Clutch Model Implementation After a numerical friction model was selected and tailored to the specific case of simulating PTER, it was necessary to implement it. The goal of the clutch model is to calculate generated clutch torques given input signals and the state of the system. The Karnopp model will yield this information, as the frictional force represented in the model is actually the generated clutch torque. Since the Karnopp model contains two distinct modes (stick and slip), there will be two different methods for calculating an arbitrary generated clutch torque, dependent on the clutch s dynamic mode. The code determines the dynamic mode of each clutch based on its relative plate velocity ω. If ω is within the range [ δω, δω] then it is considered to be in the stick mode, otherwise it is in the slip mode. If a clutch has zero input current, its reaction torque is assumed to be zero, regardless of its dynamic mode. This was done to greatly simplify the model code, and is a valid assumption since the zero-current torque generated by PTER s clutches is very nearly zero. In the slip mode, torque calculation is straightforward. According to the modified Karnopp model equation 8, the reaction torque is dependent only on the input current furnished to the clutch and the relative clutch plate velocity. A dynamic friction model for the clutches is used to compute the reaction torque. For solely friction model validation purposes, a linear model from zero to maximum rated clutch torque was used. Torque calculation for the stick mode is more involved. In this case, the modified 2

33 Case # Clutch Mode Clutch 2 Mode Clutch 3 Mode Clutch 4 Mode slip/na slip/na slip/na slip/na 2. stick slip/na slip/na slip/na 2.2 slip/na stick slip/na slip/na 3. slip/na slip/na stick slip/na 3.2 slip/na slip/na slip/na stick 4. stick NA stick NA 4.2 stick NA NA stick 4.3 NA stick stick NA 4.4 NA stick NA stick 5 stick stick NA NA 6 NA NA stick stick 7 stick stick stick stick Table : The 2 Possible Modes of PTER (N/A = not applied) Karnopp model equation 8, shows that the reaction torque is dependent on the input current furnished to the clutch and the net external torque applied to the clutch. It is therefore clear that the net external torque applied to the clutch must be computed. In order to do this, the equations of motion of the system are solved under special circumstances depending on the specific system state. It is assumed that if a clutch is in the stick mode, it will remain immobile until the net external torque applied to it exceeds its static breakaway level. Given this assumption, the equations of motion can be simplified and solved for the clutch torques required to keep the static clutches in the stick mode. This concept is made more clear by example below. The net torque on each link is the summation of clutch reaction torques and input forces transformed into torques: τ A = τ + τ 3 + τ 4 + τ A,ext (9) τ B = τ 2 τ 3 + τ 4 + τ B,ext (2) Therefore, the equations of motion of the system (Equation 5) can be written as 2

34 follows: M M 2 M 2 M 22 θ A θ B + V V 2 = τ + τ 3 + τ 4 + τ A,ext τ 2 τ 3 + τ 4 + τ B,ext (2) where the M xy values represent PTER s inertial matrix, θ x is the angular acceleration for link x, and the V x values account for velocity-dependent coupling effects. There are twelve possible dynamic modes that PTER can be in (2 4 possible from the Karnopp model with four surfaces, minus four unachievable states due to the fact that the system is overactuated, such as clutch slipping and clutches 2, 3, and 4 sticking.) These possible modes are listed in Table. We will consider Case 2. to furnish an example. In Case 2., clutch is in the stick mode and all others are in the slip mode. The calculation of the generated clutch torques for clutches 2, 3, and 4 is straightforward as discussed above. The torque required of clutch to keep itself static is solved by setting its relative angular acceleration to zero and solving the equations of motion for torque. Otherwise, if θ A = (22) then the equations of motion yield τ = ( ) M 2 τ2 M 22 ( M 2 M 22 + ) τ 3 + ( M 2 M 22 ) τ 4 + ( ) (23) M 2 τinb M 22 τ ina + V If the computed τ is below the breakaway torque for clutch, then the code returns τ for the generated torque and the clutch remains stuck. However, if it is above the breakaway torque, then the code returns the breakaway torque as the 22

35 generated torque and the clutch will eventually start to slip. See Appendix A for the full set of calculations required for each of the modes listed in Table. The primary limitation of this method is that it cannot be used for Case 7. Case 7 represents all situations where all velocities are zero and three or four clutches are applied. In these cases there are three or four unknown static torques to be found, but only two equations of motion. This yields a statically indeterminate system. In order to deal with Case 7, an alternate method of computing clutch torques had to be found. The method devised to deal with Case 7 was named the lumped actuator approach. In effect, the order of the system is reduced by considering not the four clutches independently, but as a lumped set acting on the two main links A and B. At the lowest level, all that the simulation requires from the clutch model is the net torque acting on each of the two links A and B due to the system of clutches. It is advantageous to compute the contribution of each clutch for informational and analysis purposes, and even necessary when clutches are slipping. However, in Case 7 the entire system is static, since more than one clutch is stuck, thus reducing PTER to a zero degree-offreedom device. The goal of the simulation at this point is to determine whether or not the system will remain at zero degrees-of-freedom and if not, what part(s) of the robot will start moving. To this end, the lumped actuator approach considers all the applied clutches as a single actuator which will supply appropriate torques to links A and B, up to a certain breakaway level, in order to keep the system fully static. At this point the question remains: how to determine whether or not the lumped set of clutches is capable of keeping the system static, and if not, in what fashion will the robot start to move? Case 7 is divided into five subcases, as shown in Table 2. The implementation of 23

36 Case # Clutch State Clutch 2 State Clutch 3 State Clutch 4 State 7. applied applied applied free 7.2 applied applied free applied 7.3 applied free applied applied 7.4 free applied applied applied 7.5 applied applied applied applied Table 2: Case 7 Subcases the lumped actuator approach for subcase 7. will be explained, with understanding that the other cases are treated in a similar fashion. For Case 7., clutch 4 is free and need not be considered. Several special cases of the equations of motion are defined with the intent of putting the equations of motion into a solvable form. Three cases are defined, each with one of the applied clutch torques set to its breakaway level. The torques required of the remaining two clutches to keep the system fully static can then be computed. The following equations are the results of the solved equations of motion for each of these special cases for Case 7.. τ 3 = τ ina τ,breakaway (24) τ 2 = τ inb + τ 3 (25) τ 23 = τ 2,breakaway + τ inb (26) τ 2 = τ 23 τ ina (27) τ 3 = τ ina τ 3,breakaway (28) τ 32 = τ inb + τ 3,breakaway (29) where τ xy is defined as the torque required of clutch y to keep the system static if clutch x is at its breakaway level. It is assumed that if the system of actuators is able to keep the system fully static, 24

37 that is, there exists any feasible combination of clutch torques to this end, then the system will remain fully static. Therefore if any of the three cases listed above yields both resultant torques below their breakaway levels, then the system will remain completely static. It is important to realize that when this approach is used, the clutch torques returned by the model have no physical meaning whatsoever. They do not represent the actual torques produced by the clutches. What they do represent is a means by which the two correct net link torques can be computed. With the lumped actuator approach, the net link torques will be correct, while the component clutch torques which comprise the net torques will not. In the event that one of the above special cases is not valid, it is assumed that the system will transition to a dynamic state. Now the question is how many clutches will slip, and which ones? If none of the six torques computed above are below their respective breakaway torques, it is assumed that every clutch will start to slip. In this case, the model outputs the breakaway torque of each clutch as their generated torques. If at least one of the six computed torques is below its breakaway level, then the previous timestep s torques for each clutch are used to determine which two clutches will slip. The two clutches with previous torque values closest to breakaway are the ones that will slip. When the two clutches to slip have been identified, the breakaway torques for those clutches are output by the model, and the torque for the third clutch will be computed through the now-solvable equations of motion, similar to the solutions of Cases 2 and 3. Of course, it is obvious to ask whether or not using the previous timestep s clutch values is a valid method, since it has been established that these values have no physical significance in and of themselves. However, if the system is fully static and external forces are rising to the point of dynamic transition, it is apparent that 25

38 eventually only one of the special cases will produce a satisfactory condition, i.e. both computed torques are below breakaway levels. In this case, even though the values of the computed clutch torques are not valid, the general magnitudes are; one clutch will be at or near breakaway and the system will slip when a second reaches breakaway. It can be assumed that the two clutches closest to their breakaway values will ultimately slip. In the case that the timesteps are very large, and there are two or more valid cases carried over from the previous timestep, it is impossible to guess which clutches will slip, and one guess is as good as any. As stated above, the implementation of the lumped actuator approach for the other subcases of Case 7 are similar. The exception is subcase 7.5, in which six special cases are needed instead of three, since two clutch torques must be defined in order to render the equations of motion solvable. After the definition of these six special cases, the logic of the solution is the same Dynamic Simulation Figure 7 is a diagram of the dynamic simulation portion of SimPTER. This block computes PTER s full dynamic state the angular position, velocity, and acceleration of links A and B. The inputs to the block are the net torques on each link, calculated from the input forces and the generated clutch torques, the latter being supplied by the clutch model. The block first uses the equations of motion (see Equation 5) to compute the angular accelerations of each link as follows: θ A θ B = M M 2 M 2 M 22 τ A τ B V V 2 (3) 26

39 2 Position 2 3 Vel Position Demux Ang V 4 Vel 2 Net Torque 2 Net Torque 2 MATLAB Mux Function torque >acceleration Position(2)/ Velocity(2)/ Net Torque(2) /s Demux Velocity Demux Integrator dv Karnopp differential velocity th2dot Angular Acceleration Mux Mux MATLAB Mux Function Demux Mux6 Velocity Limiter Limited Vel(2)/ Brake state byte locked thdot_unmod State of brakes Pre limited (locked/unlocked) Angular Velocity Demux 5 Pre Ang V Pre Vel 6 Pre Vel 2 /s Position Integrator thdot Post limited Angular Velocity Demux Ang Position th Angular Position MATLAB Function angles >cartesian Schwarzenegger Demux <= STOP Demux Relational Stop Simulation Operator 26 Simulation x limit position Cartesian Tip Position MATLAB Function check for singularities sing_errors singularities STOP Stop Simulation Figure 7: SimPTER Dynamic Simulation Block These are then integrated to obtain angular velocities of each link, which are again integrated to obtain angular position. Prior to integration the angular velocities are limited such that a value of zero is supplied to the position integrator if it is within the Karnopp range [ δω, δω]. A Runge-Kutta integration routine using the (4,5) Dormand-Prince pair is used for numerical integration. 3.3 Clutch Model Validation Once the new clutch model was implemented, it was necessary to perform some tests to validate its performance. It was desired to see if the clutches indeed exhibited proper static and dynamic behavior, and to examine the transition between the two states. To this end, a set of reduced-order tests were constructed. In these tests, the simulation is configured to apply a slight signal to one clutch and a ramped input force acting solely on that clutch. Initially, all other clutches are set to zero. Figure 8 shows the result of the test for clutch. From the plot of 27

40 3 25 SimPTER Endpoint Position Brake Generated Torque (in-lbs) Brake 2 Generated Torque (in-lbs) Y (in) 5 5 Constant DOF Actual Path 2 3 Brake 3 Generated Torque (in-lbs) Brake 4 Generated Torque (in-lbs) X (in) Figure 8: Reduced Order Test - Clutch Y (in) SimPTER Endpoint Position Brake Generated Torque (in-lbs) Brake 3 Generated Torque (in-lbs).5 Brake 2 Generated Torque (in-lbs) Brake 4 Generated Torque (in-lbs) X (in) Figure 9: Reduced Order Test - Clutch with Clutch 2 Locked generated torque for clutch, it can be seen that the generated torque is ramping up to compensate for the ramped input force, and clutch remains stuck. When the input torque to clutch exceeds its breakaway torque (65 in-lbs in this case), then the clutch slips and transitions to the slip mode. Since the dynamic friction model used for these tests is independent of clutch plate velocity, the model outputs a constant generated torque while the clutch is slipping. In the endpoint plot, the dotted line represents the path that PTER s tip would take if only clutch was moving. It does not exactly follow the path in this case due to the velocity coupling terms V x in the equations of motion causing link B to move as well. 28

41 Another test was performed on clutch with the same parameters, except now clutch 2 is supplied with a maximum input signal. This should eliminate the movement of link B after clutch starts to slip as evidenced in the above test. The results are in Figure 9, and it can be seen that the endpoint now follows exactly the single degree-of-freedom line. In this case, clutch 2 generates torque to counteract the influence of the V x terms, and keeps link B static. The same tests were performed on the other clutches, all with similar results. See Appendix B for the results of these tests. 3.4 Required Changes to SimPTER At this point, it has been shown that a dynamic simulation of PTER with a valid actuator model has been developed. In order to use SimPTER to make judgments about proposed changes to the physical system, accurate models for the clutch dynamics and static and dynamic friction forces must be added. 29

42 Chapter 4 PTER - Actuator Identification and Modeling 4. Motivation for System Identification After the framework dynamic clutch model had been fully implemented in the dynamic simulation of PTER, it was necessary to add models for relevant physical phenomena in order to make the simulation as accurate as possible. Prior to this point, very rudimentary models relating generated dynamic clutch torque and breakaway torque to input voltage were available. Data on the first order time constant of the clutches was inconclusive. Due to these factors, it was decided that more accurate information was required, and to perform system identification tests on PTER s clutches. 4.2 Previous Testing Gomes performed several tests on the clutches in an attempt to characterize the friction and dynamic properties. [] These tests were performed with the clutches inplace in order to avoid disassembly of PTER and the construction of testing hardware. Two types of tests were performed by Gomes a breakaway torque test for each clutch and dynamic response tests of a single clutch. The breakaway torque tests 3

43 consisted of setting one of the clutches at a constant input voltage and applying force to the tip of PTER until the locked clutch slipped. The input force at the onset of slip was transformed into a set of link torques, and the breakaway torque for the given clutch was computed. The dynamic response tests consisted of a user moving the tip of PTER with no clutches applied and applying a step input voltage to the clutch. The user attempted to keep the applicable link moving at a constant velocity. Again, generated clutch torque was computed by transforming applied tip force into applied link torques. The time response of the clutch torque was then used to obtain a first-order time constant for the system. A time delay was also noted in the clutch response. Frequency response analysis was also performed by feeding a biased swept sinewave voltage to a specific clutch and again attempting to move the particular link at a constant velocity. Gomes obtained widely differing values for the time constant in the range of.25 to 2.5 seconds from the step response and frequency response tests. It is difficult to determine a valid time constant from this data for use in the simulation. For the first implementation of SimPTER, the existing model for breakaway torque was used. Several time constants were utilized within the range identified by Gomes. Models for generated dynamic torque were created by extracting the torque-to-voltage model from the PTER control code and inverting it to obtain a voltage-to-torque model. 4.3 Initial In-Place Testing At this point, better data was needed to characterize the behavior of the clutches. It was decided to perform some additional in-place tests on a clutch for the primary 3