Control of Legged Robotic Systems: Substantiation of Gait Design, Multi-Modal Behaviors, and Dynamic Scaling Theory in Practice Daniel J

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2018 Control of Legged Robotic Systems: Substantiation of Gait Design, Multi-Modal Behaviors, and Dynamic Scaling Theory in Practice Daniel J. Blackman

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COLLEGE OF ENGINEERING

CONTROL OF LEGGED ROBOTIC SYSTEMS: SUBSTANTIATION OF GAIT DESIGN,

MULTI-MODAL BEHAVIORS, AND DYNAMIC SCALING THEORY IN PRACTICE

DANIEL J. BLACKMAN

A Thesis submitted to the Department of Mechanical Engineering in partial fulﬁllment of the requirements for the degree of Master of Science

2018

Jonathan Clark Professor Directing Thesis

William Oates Committee Member

Carl A. Moore Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the thesis has been approved in accordance with university requirements.

ii TABLE OF CONTENTS

List of Tables...... v List of Figures...... vi Abstract...... ix

1 Introduction 1

2 Background 4 2.1 Natural Legged Locomotion...... 4 2.2 Simulation and Robotics...... 4 2.2.1 Bioinspiration versus Biomimetics...... 5 2.2.2 Gait Controller Design for Multi-legged Systems...... 6 2.2.3 Dynamic Scaling...... 7

3 Simulation and Experimental Design8 3.1 Introducing Minitaur...... 8 3.1.1 Initial Control Strategies...... 8 3.1.2 Isolating Sagittal Plane Dynamics...... 10 3.2 Simulation Modeling...... 12 3.2.1 SLIP Modiﬁcation...... 12 3.2.2 Hopping...... 14

4 Gait Development on Minitaur 15 4.1 Experimental Setup for Gait Analysis...... 15 4.2 Minitaur Crawl Gait Results...... 15 4.3 Controller Design Modiﬁcation...... 16 4.4 Updated System Running...... 17 4.5 Walk to Trot...... 18

5 Running and Jumping Behavior 20 5.1 Running...... 20 5.1.1 Simulation of 5-bar SLIP...... 20 5.1.2 Experimental Validation of 5-bar SLIP...... 22 5.2 Jumping...... 23 5.3 Running and Jumping...... 25

6 A Physical Manifestation of Dynamic Similarity and Scaling 28 6.1 Designing at the Larger Scale...... 29 6.2 Characterizing Limitations: System Identiﬁcation...... 29

iii 7 Conclusions 38

References...... 40 Biographical Sketch...... 44

iv LIST OF TABLES

3.1 Physical Parameters of the Minitaur Robotic Platform...... 8

5.1 Single-leg Running Experimental Results of Stability Analysis (this table is reproduced from [9] c 2017 IEEE)...... 22

6.1 Scaling Parameters Relating the UPenn Thumper and ARL Hopper...... 34

v LIST OF FIGURES

1.1 The early stages of the quadrupedal robotic platform Minitaur (left; image taken by author) and a single Minitaur leg attached to a mechanical boom system for constrained sagittal plane dynamic analysis (right). This image is reproduced from [9] c 2017 IEEE...... 2

2.1 This ﬁgure shows demonstrative images for distinguishing the diﬀerence between bioinspired and biomimetic engineering design...... 6

3.1 A single Minitaur leg (a) is depicted here with deﬁnitions of the angle parameters and link lengths used to determine leg length and touchdown angle. The trajectory is performed in the motor space (b) with the rotation of both motors together during stance and in opposition to provide toe liftoﬀ and touchdown...... 9

3.2 Robotic platforms designed to explore dynamic scaling with the larger UPenn Thumper (b) exhibiting the same kinematics but at a larger scale (αl = 1.5 and αm = 2.12)... 11

3.3 The dynamics and deﬁnitions of the SLIP model (a) and their translation to the 5-bar kinematic leg design (b, c). These ﬁgures are printed with permission [9] c 2017 IEEE. 12

4.1 Raw data from the motion capture system for velocity (a) was used ﬁltered and diﬀerentiated (b) to calculate a window of steady system velocity (c). This image is reproduced from [10]...... 16

4.2 Manipulations of the triangular trajectory include the height of toe lift (a) and the position of the apex height with respect to commanded touchdown (b). The results from an experimental analysis of the eﬀects of parameter sweep (c) of the updated control used on Minitaur demonstrate greater velocities than seen previously (Fig. 4.1d). This image is reproduced from [10]...... 18

4.3 Deﬁnitions and experimental results for walk-to-trot gait development (a) as well as a time-lapse of Minitaur performing a trot on the plywood surface (b) used through this gait study. These images are reproduced from [10]...... 19

5.1 Stability regions of running are shown here for a range of different postures (rows) and linkage ratios (columns) of the 5-bar kinematic SLIP model. The images in the left most column depict half of the symmetric linkage to demonstrate the posture of the leg with the different nominal linkage configuration angles. For the plots, the axes are non-dimensional stiffness versus angle of attack (α0 = 90 + ψ), with a secondary y-axis on the right demonstrating the torsional stiffness constant of the resultant hip ◦ ◦ ◦ spring (from Eq. 3.9). The three posture angles (θ0 = 45 , θ0 = 75 , and θ0 = 105 ) were chosen based on the physical limitations of the robot (linkage collision occurs at angles less than θ = 45o) and previous work regarding knee springs with a ratio R = 1/1. Reproduced with permission from [9] c 2017 IEEE...... 21

vi 5.2 (a) The 1D hopping model performs a single jump, starting near full compression, extending until liftoff is achieved through force balancing between the actuator output and gravity. (b) For a range of different ratios at three designated control efforts (% voltage to motors), the hop height displacement demonstrated optimal ratios of 0.54, 0.60, and 0.62 (for 50%, 75%, and 100% effort, respectively). This figure is reproduced with permission from [9] c 2017 IEEE...... 24

5.3 A single, powerful jump with the FSU MiniBoom platform begins with the leg at full compression (a) with output torque to both motors until full extension (b). The results of the hop height measured from maximum height of the boom arm are normalized for comparison between R = 1/1 and R = 1/2 (c). Reproduced with permission from [9] c 2017 IEEE...... 25

5.4 Experimental data depicting the run-to-jump results for the R = 1/2 and R = 1/1 legs. Reproduced with permission from [9] c 2017 IEEE...... 26

5.5 Photographic timelapse demonstrating the conﬁguration of the R = 1/2 compressing and subsequently clearing the obstacle with its jumping capability. Reproduced with permission from [9] c 2017 IEEE...... 27

6.1 Sketch design for Thumper platform produced by Wei-hsi Chen of the Kod*lab at University of Pennsylvania...... 29

6.2 Analysis of 1D hopping on the ARL Hopper with variation in controller damping for two distinct system masses...... 30

6.3 Multiple leg drops are performed on the ARL Boom (a) then individual drop events are isolated (b). The necessary output values are extracted including the ﬁrst two minimum peaks, the maximum peak, the period of oscillation, and the steady-state leg length...... 32

6.4 Calibration curves for stiﬀness (a) and damping (b) determined using the described fsolve method for the ARL Hopper system...... 32

6.5 A Comparison of the ARL Hopper with added increments of +0.1kg and +0.2kg demonstrates similar steady-state hop heights when scaling is applied to the one dimensional AER controller...... 33

6.6 Calibration curves for stiﬀness (a) and damping (b) determined using the described fsolve method for UPenn Thumper...... 34

6.7 Theoretical motor speed/torque curves for the T-motor U10 Plus, DYS BE8108-16, and scaled down version of the U10 Plus...... 36

6.8 Experimental analysis of a range of diﬀerent control parameters for AER control scaled steady-state hopping between the UPenn Thumper and ARL Hopper...... 37

vii 6.9 Scaled matching of two dimensional, steady-state running between the UPenn Thumper and ARL Hopper comparing apex of ﬂight phase, minimum compressed leg length, frequency of running, forward system velocity, touchdown angle, and stance angle swept. 37

viii ABSTRACT

Through limb structure and neuromuscular control, animals have demonstrated the ability to navigate obstacles and uneven terrain using a variety of different mechanisms and behaviors. Learning from the capabilities of animals, it is possible to develop robotic platforms that can aid in the study of these motions towards the production of new technologies for military, search and rescue, and medical applications. To produce these systems, it is important to first understand the underlying dynamics and design principles existent in nature that afford creatures such dexterous and agile movements. The creation of robots with legs provide a means for studying different aspects of the dynamics of legged locomotion. This includes investigations of limb coordination for gait controller design, the role of passive compliance in dynamic running, mechanical leg design and configuration for optimal energetic output, and scalability of legged systems in both simulation and through experimentation. This thesis aims to provide insight into the design and implementation of terrestrial robotic platforms with legs.

ix CHAPTER 1

INTRODUCTION

The versatility afforded by legs allows animals of all shapes and sizes to roam the earth, unencum- bered by many of the limitations that befall wheeled vehicles. Though wheels offer greater mobility over smoother surfaces, nature has clearly demonstrated the ability of legs to overcome unstructured environments with greater agility and dexterity. For this reason, legged robotic systems have been developed to study the interaction of legs over irregular terrain in both man made and natural environments with inspiration from biological systems [20, 35, 38]. These systems are generally tasked with specific goals in mind and are thus optimized to perform the desired objective of that particular machine. It is the intention that through understanding the characteristics that define motion entirely, a basis will be formed for streamlining the process of behavioral development for legged robots. Arguably, if we determine the dynamical similarities of two, legged platforms, similar in design but differing in size, it should be possible to scale the characteristic control parameters in order to develop the same behaviors on both robots. This was demonstrated in simulation previously, through the derivation of the dynamic scaling laws [32] which allow direct scaling of control variables through dimensional analysis. Though this study successfully provided a means of controller scaling through the use of free variable ratios, it failed to demonstrate a substantial realization of this theory through implementation on a physical platform though an example was noted. RHex [18] and Edubot [17] are two hexapod robots of similar design that required time consuming op- timizations of control and mechanics to produce the fast, agile gaits achieved by each respective platform. Repetition of this process could have been avoided through measurement of certain physical attributes for comparison and ultimately the application of dynamic scaling theory. These attributes include such features like mechanical damping, power and actuation limitations, and friction which prove to be of significant importance in the realm of scaling and require adequate evaluation.

1 (a) Minitaur (b) FSU MiniBoom

Figure 1.1: The early stages of the quadrupedal robotic platform Minitaur (left; image taken by author) and a single Minitaur leg attached to a mechanical boom system for constrained sagittal plane dynamic analysis (right). This image is reproduced from [9] c 2017 IEEE.

To understand these features, it is important to analyze the dynamics involved in the development of gaits on a legged system. The multiple degrees of freedom of a biped, quadruped, and hexapod influence the body dynamics and their interaction with the ground and obstacles in a variety of different ways. Understanding the compliance of the system, stability involved in foot placement, and adaptability to different surfaces and obstructions is pivotal to encapsulating the important components necessary for controller scaling. For this reason, it is helpful to design gaits in a way that provides intuitive control variables that have physical value to the systems interactions. Single leg platforms can also provide a significant amount of information regarding these features since multi-leg dynamics are well represented through the dynamics of a single-leg or bipedal system [36]. With this simplification in mind, low-order models can be utilized in order to properly capture the general concepts necessary to understanding legged locomotion and how it is influenced by compliance within the system [11, 19]. A key feature of the dynamics of locomotion is the capacity to perform energetic tasks such as jumping. For this reason, the robot Minitaur [25] uses a symmetric, closed-loop linkage for

2 its leg design that allows it to perform explosive jumps that reach impressive hop heights [26]. Comparisons have been made between the energetics of machines like Minitaur and jumping robots such as the University of California Berkley’s SALTO [21] This combined with the high frequency response of actuation present on Minitaur results in the capability of producing fast running gaits, with speeds reaching up to 2.4m/s [12]. Understanding these key behaviors through robotic control forms the foundation for developing legged systems of all shapes and sizes. Once a legged system is well understood, the dynamic scaling laws can be applied to perform direct controller mapping to similar robotic platforms. In this thesis, the initial gait development process performed on Minitaur is explained to highlight the lessons learned about multi-legged locomotion. Then, a control method is developed for a single Minitaur leg on a boom to explore the running stability of the mechanical design and transitioning for obstacle negotiation. Finally, this single leg platform is utilized in conjunction with a similar, but 1.5 times larger, system at the University of Pennsylvania’s Kodlab to provide a physical representation of dynamic scaling laws and evaluate the aspects of this process that require special attention when dealing with actual hardware systems under real-world constraints. The speciﬁc contributions of this thesis include:

1. Demonstration of a clock-based gait on a quadrupedal robotic platform that utilizes unique kinematics for its leg design and producing up to 1.4m/s forward velocity

2. Analysis of the stability regions for running with symmetric 5-bar legs and the application toward coupled, running and jumping behaviors to overcome obstacles of nominal leg length height

3. A physical representation of dynamic scaling and similarity theory as well as an address of the challenges involved

3 CHAPTER 2

BACKGROUND

2.1 Natural Legged Locomotion

In nature, animals have demonstrated the unique ability to navigate the rough terrain, both man-made and natural, that exist in the world. With great agility and efficiency, organisms are able to overcome obstacles, outrun predators, and traverse uneven surfaces, with a level of finesse that remains unrivaled by modern machines. For example, the cheetah is capable of reaching speeds up to 29m/s (or ≈ 65mph)[41] to catch gazelle and impalas in sub-Saharan Africa through the use of a flexible spine [16]. Birds of all sizes (like the quail or ostrich) are able to account for perturbations in terrain elevation while running through biasing the passive compliance of their legs at liftoff and touchdown using reflex/preflex movements (which is notably done for efficiency and self preservation) [8]. Some animals, such as the galago (otherwise known as bush baby) are capable of performing long, steady-state jumps to traverse dense vegetation quickly, reaching greater heights among the tree tops to avoid predators [4]. These aspects of speed, efficiency, energetics, and stability optimization allow animals to navigate in unstructured environments.

2.2 Simulation and Robotics

Though this gap in technology exists, efforts are being made to both bridge the divide as well as understand the principles that allow animals to move with such dexterity in unstructured environments. One approach to this analysis is through the use of physics-based simulation models that try to capture the underlying dynamics of animal mechanics and ground interactions. For example, walking and running are each simplified to a sagittal plane motion through the use of the low-order dynamics models including the Double Inverted Pendulum Walking Model [13, 29] and the Spring-Loaded Inverted Pendulum (SLIP) Model [11]. Though these models provide a general template for analyzing the dynamics of legged locomotion, they unfortunately lack certain key aspects of the specific mechanisms animals use to increase stability and efficiency. For this reason, studies have been done to explore different principles observed in nature through modification of

4 these models. For example, one study replaced the linear spring present in the SLIP model by re- designing the pendulum to be a two-segment leg in order to capture the torsional dynamics exhibited by a standard knee actuated conﬁguration [37]. Another modiﬁcation included the combination of the above SLIP and walking model through the addition of springs to a double inverted pendulum [19], demonstrating the contribution of leg compliance to stable walking of a legged system.

2.2.1 Bioinspiration versus Biomimetics

Another method involves the use of biomimetic or bioinspired robotic platforms that aim to tackle the complex challenges of design and control that nature has already exhibited while providing all of the physical hindrances that exist in the real world. The realm of biomimetics involves production of robotic systems that closely mimic the movements and mechanical design of animals in order to better understand how they are able to perform the tasks that they do. For example, ornithopters are built to study the aerodynamics of flight produced through flapping of the wings as is done by birds [30]. By contrast, bioinspiration more generally looks to nature for clues for optimal control and design but focuses more on engineering principles since there are structural aspects of organisms that have to yet to be directly matched by technology. The counter example to an ornithopter would be a quadcopter, which uses a slightly different method for flight but can express aspects of biological control through vision-based formation flying and the incorporation of biological design characteristics into existent engineering solutions. [14, 15]. Figure 2.1 provides a comparison through visual examples of bioinspiration and biomimetic engineering. Through bioinspiration, roboticists have explored legged locomotion with the goals of understanding and optimizing speed, dexterity, efficiency, and stability. At this point, efficiency has been best expressed through exploiting the underlying dynamics described through the walking and running simulation models. Through coupling design and controller optimization, the Biorobotics and Locomotion Lab at Cornell University produced the human scaled walking robot, Ranger, that is capable of dynamically walking up to 65km on a single charge of its battery [6]. At MIT, the Biomimetic Robotics Lab focused on tackling untethered speed through their production of the Cheetah robot, which was capable of reaching speeds of up to 22km/hr and transitioning from a trotting to galloping gait [40]. With these fast and efficient platforms in mind, there’s still a lot to be said about the energetics involved in locomotion. Some would argue that the ability to produce

5 Figure 2.1: This ﬁgure shows demonstrative images for distinguishing the diﬀerence between bioinspired and biomimetic engineering design.

explosive jumping capabilities at high frequencies like the University of California Berkley campus’ SALTO robot is an ideal indication of system agility and performance [21].

2.2.2 Gait Controller Design for Multi-legged Systems

The development of gait control is a key aspect in the field of legged locomotion. Often, control schemes are developed using methods inspired by nature that incorporate neural networks or adaptive control in order to overcome real world surface interactions and obstacles in an efficient manner [7, 24, 43, 44]. Though these control schemes are interesting, they offer little intuition into the mechanics and design that allows for optimal performance of robotic systems through their ”black box” approach to determining gait solutions. An alternative to these approaches involves the use of clock-based slaving of leg coordination which was a popular method adopted on iSprawl at Stanford University and RHex at the University of Pennsylvania [28, 38]. Additionally it is advantageous to draw foot trajectories (in the case of multiple degree of freedom legs) through either geometrical designation and setting of way-points [27], matching of the kinematic motion primitives of animals (such as horses) [34], or impedance-based proprioceptive feedback control with

6 mathematical mapping of way-points [23]. Combining these features with common leg offset timing expressed by quadrupedal species [22, 39] provides an intuitive control method with representative parameters through variables such as leg frequency, foot way-points, and limb timing offset to make the process of analysis straight forward for getting a system up and running, quickly and efficiently.

2.2.3 Dynamic Scaling

Having intuitive control methods is useful but there are far more advance control schemes available with the growing interest in legged robots. For this reason, it is of signiﬁcant interest to start designing a streamline protocol for porting these controllers between legged systems of varying size and mass. Previous work simulated the process of dynamic scaling as applied to dynamic legged systems [32], highlighting the main assumptions that can be manipulated in the case of robotics. The scaling factors derived in this work can easily be deduced through dimensional analysis of the parameter in question (i.e. the scaling factor for a unit of velocity is αl/αt since the factor for length is αl and for time it is αt). Knowing this and assuming that gravity experienced at every scale is going to be constant,

2 αg = αl/αt = 1 (2.1) √ which demonstrates that αt = αl. A secondary constraint discussed demonstrates that if the same materials are used in the geometric design of a larger system, then it can be assumed that the density is constant so,

3 αρ = αm/αl = 1 (2.2)

3 with αm = αl at every scale. In robotics however, these variables can be manipulated through material selection and plane- tary exploration or buoyancy manipulation. For example, a robot on the scale of 1gram could be made of paper materials while a robot at the 100gram scale would benefit from structures built from titanium or carbon fiber. Through these design choices, robotic platforms can be built at different scales with controller optimization done at the smaller, less costly size to avoid significant issues.

7 CHAPTER 3

SIMULATION AND EXPERIMENTAL DESIGN

3.1 Introducing Minitaur

Collaboration with the Kod*lab at the University of Pennsylvania through the US Army funded RCTA program led to the development of a quadrupedal platform named Minitaur [25]. Minitaur is a unique robot due to the mechanical design of its legs. For each leg, two motors are located coaxial to one another with space in between for the closed-chain, 5-bar linkage. The 5-bar linkage is essentially a 4-bar system in the sagittal plane with an additional bar connecting the two actuated linkages as ground (which is demonstrated in previous work on the mechanics of this closed-loop design [5]). The motors are directly attached to the cranks of the linkage, allowing for high bandwidth of proprioception. This means that rather than incorporating sensors to detect and measure forces, estimations can be made using the Jacobian and motor torques (dependent on electrical current draw) to determine and measure impact. The high torque, T-motor U8 [2] is used on this system with electronics developed by electrical engineering students at the University of Pennsylvania.

Table 3.1: Physical Parameters of the Minitaur Robotic Platform

Minitaur Mechanical Parameters Parameter Value Mass 5.15 kg Length 45.72 cm Width 30.47 cm Primary Link (L1) 10 cm Secondary Link (L2) 20 cm Toe Extension 2.5 cm

3.1.1 Initial Control Strategies

To produce a forward motion on the Minitaur quadrupedal platform, work began through utilizing the motor space and an open loop control formulation. This was done through proportional

8 (a) CAD Model of Minitaur Leg (b) Foot Trajectory Mapping

Figure 3.1: A single Minitaur leg (a) is depicted here with deﬁnitions of the angle parameters and link lengths used to determine leg length and touchdown angle. The trajectory is performed in the motor space (b) with the rotation of both motors together during stance and in opposition to provide toe liftoﬀ and touchdown.

control of the motors and time syncopation of the leg motions to produce a desirable foot trajectory. The trajectory is defined by setting a predefined value for the ratio between stance and flight time for the period of circulation, or a duty factor. This separates the trajectory of a single leg into two main phases, the stance phase and the flight phase. During the stance phase, the limb is in contact with the ground while in the flight phase the leg is lifted and resets to the subsequent touchdown angle. The duty factor manipulates the which phase is shorter, dependent on the set frequency of footpath completion, and defines the overall gait timing.

The stance phase begins with motor angles θ1 and θ2 (defined in Fig. 3.1a) initially at a defined touchdown angle (TD) as shown in Fig. 3.1b. During the stance value the motors rotate from the position of touchdown (TD) to liftoff (LO) based on the set duty factor (DF) and frequency. The angle between TD and LO is defined as the sweep angle, ψs, and determines the stride length of the system. With values θ1 = 90ψs/2 and θ2 = 90ψs/2 defining touchdown, the rotation of the leg begins with a rate defined as

9 ψ R = s (3.1) 1 DF At LO, the toe lifts off the ground through rotational opposition of the motors and defined by ◦ a ground clearance factor gcl which is done to avoid the kinematic singularity (θ1 = θ2 = 0 or ◦ θ1 = θ2 = 180 ). Two different rates were chosen through a preliminary Matlab simulation analysis to ensure that toe would be centered after the first half of the flight phase by

g ψ R = cl s (3.2) 2 1 − DF/2

(1 + g )ψ R = cl s (3.3) 3 1 − DF/2 The symmetry of this leg trajectory mapping allows the touchdown angle to be reset through exchanging the rates and directions between the motors. The results in the lowering of the foot to the touchdown angle, allowing continuation of the cycle for locomotion. Between the legs of the quadruped, the a leg pattern was defined with a quarter time offset between each leg to produce a crawl gait. This order of this gait pattern involved initiation of the front right leg first, followed by the hind left, front left, and hind right (FR-HL-FL-HR) based on previous studies of quadrupedal animal locomotion [39, 22]. This formulation formed the basis for the gait utilized by Minitaur to perform walking and trotting.

3.1.2 Isolating Sagittal Plane Dynamics

To further analyze the properties of single leg dynamics of Minitaur, it is valuable to isolate a single leg and constrain it to sagittal plane locomotion. An experimental platform was developed at Florida State University for just this purpose, the FSU MiniBoom 1.1b. This boom allowed for adjustable height (to match nominal leg length of the 5-bar linkage) and restricts the robot to a circular path with a radius of 1.15m. Encoders (Accu-coder model 15s encoders; quadrature phase) actively track the position of the boom arm in two dimensions during the performance of locomotion behaviors, providing 0.14mm resolution vertically and 0.044mm rotationally. These encoders are run by a myRIO at 100Hz in tandem with high speed video (240fps) for veriﬁcation of performance. A spherical foot, encapsulated in elastomer is added to provide physical compliance for system preservation over time.

10 (a) ARL Hopper (b) UPenn Thumper

Figure 3.2: Robotic platforms designed to explore dynamic scaling with the larger UPenn Thumper (b) exhibiting the same kinematics but at a larger scale (αl = 1.5 and αm = 2.12).

Additional systems modeled on the same concept as the FSU MiniBoom were designed by the Army Research Laboratory (ARL) and University of Pennsylvania Kod*lab (Fig. 3.2). These systems were constructed with the speciﬁc intent of exploring the concept of dynamic scaling since the UPenn Thumper utilizes a U10 T-motor (as opposed to the U8 on Minitaur) [2] and has a geometrically 1.5 times larger leg than the Minitaur scale of FSU MiniBoom and ARL Hopper. All three systems utilize the 5-bar mechanical design demonstrated by Minitaur and are driven using the same electronics [1, 10, 42]. These electronics were developed by the team at the University of Pennsylvania and consist of a STM32F303 mixed-signal MCU with a ARM-Cortex M4 core capable of running at 72 MHz.

Controller Design. In this two dimensional running domain, two main controllers are used to produce dynamic locomotion on the system. The first is triggered when the leg reaches maximum compression (and begins to deflect) and its hip has rotated beyond the center of pressure (contact of the foot with the ground). With this event, the motors are commanded to perform an open loop torque output of a specified percentage of effort until the leg reaches the defined nominal length. This controller is utilized for both steady-state running and single hopping experiments. The

11 second controller, called active energy removal (AER), provides a sinusoidal oscillation of nominal leg length that begins shortening the leg upon impact and then lengthening it until liftoﬀ [31]. This control works through set parameters of driving frequency, amplitude of oscillation (as a percentage of leg length), and timing oﬀset (between the touchdown event and the sinusoid). This controller is used in running as well as one dimensional steady-state hopping.

3.2 Simulation Modeling 3.2.1 SLIP Modiﬁcation

(a) Standard SLIP Model (b) Modiﬁed 5-bar SLIP (c) Labeled Robot

Figure 3.3: The dynamics and deﬁnitions of the SLIP model (a) and their translation to the 5-bar kinematic leg design (b, c). These ﬁgures are printed with permission [9] c 2017 IEEE.

With the development of single leg platforms to study running dynamics, simulation becomes necessary to provide a starting point for analyzing and optimizing control strategies. For the particular leg design utilized by Minitaur, modification is made to the standard 2D SLIP model (Fig. 3.3). Setting configuration angle of the motor cranks and commanding a proportional gain, the virtual leg exhibits nonlinear compliance between the hip and the toe, reminiscent of the effect analyzed for a two-segment leg with an incorporated knee spring [37]. Utilizing the Lagrangian Method for calculating the equations of motion, the system dynamics are defined as

κ dθ ζ¨ = ζψ˙2 − gcos(ψ) + (θ(ζ ) − θ(ζ)) (3.4) m dζ 0

12 2ζ˙ψ˙ g ψ¨ = − + sin(ψ) (3.5) ζ ζ in which ζ is the dynamic leg length, ψ is the angle deﬁned from the vertical center of the leg (parallel to the direction of gravity), g is the gravitational acceleration, κ is the torsional spring stiﬀness, and m is the point mass at the hip. The touchdown event is detected when the vertical component of the projected nominal leg length is equal to the height of the center of mass (COM). This can be expressed as yTD =

ζ0cos(ψTD) with nominal leg length expressed via the kinematics of the leg through the hip angle, θ via the relation

q 2 2 2 ζ0 = λ2 − λ1 + (λ1cos(θ0)) + λ1cos(θ0) (3.6) with pre-determined link length values for primary, λ1, and secondary, λ2, leg segments (previously referred to on the robot as L1 and L2).

Reference Stiffness. To study the effects of the nonlinear stiffness realized through the kinematics and incorporation of a torsional spring, a referential measure of stiffness is determined. This is done through assuming a non-dimensional linear reference stiffness at 10% leg compression [37]. In order to do this, the Jacobian is calculated through taking the derivative of the function describing leg length with respect to the inner linkage half angle (Eq. 3.6),

dζ dθ ζ˙ = θ˙ → F = τ (3.7) dθ dζ Using this expression the relation can be drawn between the force and torque of a spring mass system with,

dθ k∆x = κ∆θ (3.8) dζ Solving for the derivative and re-arranging the equation to account for a 10% length compression, the torsional stiﬀness required to have a comparable starting point for nominal leg stiﬀness,

˜ ∆ζ10% dζ κ = k10% (3.9) (θ0 − θ10%) dθ

13 where, k10% is the relative linear spring stiﬀness, ∆ζ10% is the length of leg compression (0.1ζ0),

θ0 is the nominal inner linkage half angle, θ10% is the inner linkage half angle at 10% length compression, and dζ/dθ is the one dimensional Jacobian (3.7). For the purposes of comparison, ˜ this value is normalized to k10% = k10%ζ0/mg.

Normalization of Linkages. To provide comparable models in simulation to understanding the underlying dynamics and avoid diﬀerences in energy required for adjustments of leg length, the linkages were adjusted through normalization of the nominal leg length and utilization of the ratio between primary and secondary with the ratio, R = λ1/λ2 and Eq. 3.6 to solve for linkage lengths,

ζ λ1 = (3.10) p1/R2 − sin(θ)2 + cos(θ) Energy Input and Stability. The input energy was maintained through setting an initial launch height equal to the nominal leg length set and velocity for each value of touchdown and nondimensional stiﬀness. The leg would then attempt to complete 25 steps and if it succeeded, a ﬁxed point analysis of apex height perturbation was performed with a resolution of 10−6m to determine steady-state stability through

dyi+1 < 1 (3.11) dyi

In which dyi+1 is the diﬀerence in the outputs between the perturbed and unperturbed step and dyi is the diﬀerence in the inputs (i.e. the perturbation).

3.2.2 Hopping

For 1D hopping, the running model was modiﬁed to include an input torque. The input torque was deﬁned using a standard motor model with parameters for the T-Motor U8 brushless DC motor used on Minitaur,

τmax τ = τmax − θ˙ (3.12) ωNL with a max torque, τmax, of 7.66 Nm and no-load speed, ωNL, of 168 rad/s (which are calculated form the motors velocity constant, Kv, and resistance, Rm provided) for the FSU MiniBoom.

14 CHAPTER 4

GAIT DEVELOPMENT ON MINITAUR

4.1 Experimental Setup for Gait Analysis

To analyze the control described in Section 3.1.1, experiments were run involving parameter sweeps of frequency, duty factor, and toe direction on the Minitaur platform. This was done in a 10-camera Vicon 1.3 Motion Capture System setup on plywood (for increased surface friction) and the robot was run on a 4-cell lithium polymer battery with a randomized order of trials to reduce the effect of physical and electrical inconsistencies on the outcomes. The open loop control was run with an analysis of overall system velocity. For each experiment, Minitaur started at rest and would walk the prescribed gait with the parameters of interest for that particular run. The direction of travel was defined by the toe, in which the forward direction is shown in Figure 3.1a while the backward direction involves moving in the opposite direction with respect to the toe. The data from an example of a single run is depicted in Figure 4.1a-c. The velocity of the system is plotted in it’s raw form to show what is initially produced from the motion capture system. This data was then smoothed using a Robust Lowess Regression smoothing algorithm with a 10% span in Matlab to eliminate the high frequency noise and differentiated to obtain the filtered acceleration. The peaks of maximum and minimum acceleration were then used to set the boundaries of a window for determining system velocity. These points signify the initial acceleration and abrupt stopping of the robot and were used to determine the window in which velocity was maintained. The window from maximum to minimum acceleration was then decreased on either side by 5% to isolate the portion of steady velocity and the mean and standard deviation were obtained.

4.2 Minitaur Crawl Gait Results

Parameter sweep of frequency, toe direction, and duty factor was performed using the analysis method described in Section 4.1. Figure 4.1d depicts these results which demonstrates several key features of the gait design for this platform. First, as intuition would suggest, increasing the driving

15 (a) Raw Velocity (b) Filtered Acceleration

Figure 4.1: Raw data from the motion capture system for velocity (a) was used ﬁltered and diﬀer- entiated (b) to calculate a window of steady system velocity (c). This image is reproduced from [10].

frequency of the gait pattern results in an increase in velocity. Second, decreasing the duty factor from 95% to 75% also produced greater speeds. Finally, the toe direction had a signiﬁcant impact on the velocity of the system, most likely due to the magnitude and direction of the resultant friction cone with respect to ground contact [3]. The maximum velocity was produced with a backward toe direction, driving frequency of 4Hz, and duty factor of 75%. This resulted in a steady speed of 0.52m/s or 1.16mph.

4.3 Controller Design Modiﬁcation

To increase system performance, the controller was re-designed to allow more intuitive manipulations toward faster, more efficient gaits. This involved specific focus on the foot path trajectories and how they were defined, through moving from a motor-space based control to Cartesian defini- tions of foot location. Additionally, the system had undergone some mechanical modifications at

16 this point through analysis of a series of issues discovered in the initial design during the initial experimental trials. Focusing on the trajectory of the control strategy, several signiﬁcant updates were incorporated into its design. First, the kinematics were calculated from the leg design (as was done for simulation studies; Eq. 3.6). The inner half angle is calculated as the average of the angles θ1 and θ2 from Figure 3.1a while the leg center angle is the resulting diﬀerence of the angles divided in half

(ψ = [θ1 − θ2]/2 from Fig. 3.1a). These polar coordinates provide a basis for translation to Cartesian coordinates via basic geometry and form the foundation for designing toe trajectories. The design of the toe trajectory for Minitaur was inspired involved the setup of deﬁned way- points [27]. For the initial shape, an acute triangle was chosen to nearly matched the initial motorspace trajectory created, in which the base would count as the stance phase and travel to and from the apex point serves as the ﬂight phase.

4.4 Updated System Running

After implementation of the aforementioned modifications to the controller and mechanical structure of Minitaur, experimental runs were performed to analyze the effect of these changes. However, since the previous study provided incite regarding frequency and duty factor, these parameters were set at 3Hz and 75% respectively (these values were chosen since they were observed to produce more stable, consistent results in the previous set of trials). In this set, the manipulation involved an analysis of manipulation of the stance posture (Fig. 4.2a) and toe trajectory apex (Fig. 4.2b), given the new dimensions of control manipulation. The results of this set are shown in Figure 4.2c. Generally speaking, it seems apparent that the motor angle configuration of 90◦ produces the fastest gaits. This demonstrates that the regime of maximum torque requirement provides greater output upon compression which has been expressed previously [26]. The apex of the triangular trajectory provides optimal results in most cases when the leg reaches beyond its intended touchdown target (the obtuse configuration or green triangle in Figure 4.2b). This control most likely provides a near-ground-speed matching affect, contributing to the stability dynamics of the system. The backward facing toe direction often elicits the greater speeds as seen before as well. This culmination of parameters results in a gait that achieves a velocity of 0.92m/s (or ≈ 2.05mph).

17 (a)

(b) (c)

Figure 4.2: Manipulations of the triangular trajectory include the height of toe lift (a) and the position of the apex height with respect to commanded touchdown (b). The results from an experimental analysis of the eﬀects of parameter sweep (c) of the updated control used on Minitaur demonstrate greater velocities than seen previously (Fig. 4.1d). This image is reproduced from [10].

4.5 Walk to Trot

Fast, stable walking on quadrupedal platforms provides the necessary speeds to move into more efficient, high speed gaits such as trotting. The transition between walking and trotting is very smooth considering the fact that walking involves a quarter offset of the period between legs and trotting is a half time offset (footfall patterns of each are demonstrated in the left portion of Figure 4.3a). A slight increase in driving velocity of 3.5Hz was applied to the successful walking gait for a skewed triangular trajectory at the normal motor angle of 90◦ with DF = 75%. This was then transitioned to a trotting gait after a set time period (ensuring steady-state walking had been achieved) at a frequency of 4.5Hz. The resulting gait reached an average velocity of approximately 1.33(±0.087)m/s with a maximum speed of ≈ 1.42m/s for a short duration before the robot ran out of space to run. It is noteworthy to point out that this is nearly three times the speed of the initial maximum velocity walking gait, demonstrating a swift timeline for controller development on a robotic system.

18 (a) Walking to Trotting: Gait Pattern and Experimental Results

(b) Photographic time-lapse of Minitaur trotting

Figure 4.3: Deﬁnitions and experimental results for walk-to-trot gait development (a) as well as a time-lapse of Minitaur performing a trot on the plywood surface (b) used through this gait study. These images are reproduced from [10].

19 CHAPTER 5

RUNNING AND JUMPING BEHAVIOR

To abstract away the contributions of leg design and control to system dynamics, it is a common practice to develop single leg simulation models (Sec. 3.2.1) and physical systems (Sec. 3.1.2). These models allow the isolation of key features that inﬂuence diﬀerent aspects of running that are necessary to produce dynamic gaits. This is especially necessary in the case of obstacle negotiation, when a robot is tasked with overcoming an object in its path while maintaining fast, stable running.

5.1 Running

In approaching the problem of obstacle negotiation, an analysis is ﬁrst done in simulation on the dynamic stability of the 5-bar leg through using the model described in Section 3.2.1.

5.1.1 Simulation of 5-bar SLIP

Figure 5.1 depicts the results of a parameter sweep of angle of attack (α0), posture configuration ˜ angle (θ), nondimensional stiffness (k10%), and linkage ratio (R) of this model. The postures examined here include an exaggerated knee below hip, θ = 45◦, knee below hip (nearly neutral), θ = 75◦, and knee above hip, θ = 105◦. Two different linkage ratios were examined between the primary and secondary linkages that make up the 5-bar leg (while maintaining symmetry in the mechanism). From these results, it can be observed that with a more upright posture (increasing θ), the system stability is greater. This is true for both ratios, depicted by the visually greater are of the sliver representing the basin of stability. Additionally, the torsional stiffness (second y-axis) required for the more upright postures is generally lower. Comparing between linkage ratios, the area of the stability region is slightly lower for ratios of R = 1/2 in each case but the torsional stiffness requirement is significantly lower. This is indicative of the energy input necessary for the motors in an experimental system since the proportional gain of the motor translates to the torsional stiffness produced at the hip. Finally, looking at the ratio of R = 1/2 affords the system to

20 Ratio = 1/1 Ratio = 1/2 o = 45 θ o = 75 θ o = 105 θ

Figure 5.1: Stability regions of running are shown here for a range of different postures (rows) and linkage ratios (columns) of the 5-bar kinematic SLIP model. The images in the left most column depict half of the symmetric linkage to demonstrate the posture of the leg with the different nominal linkage configuration angles. For the plots, the axes are non-dimensional stiffness versus angle of attack (α0 = 90 + ψ), with a secondary y-axis on the right demonstrating the torsional stiffness ◦ constant of the resultant hip spring (from Eq. 3.9). The three posture angles (θ0 = 45 , θ0 = ◦ ◦ 75 , and θ0 = 105 ) were chosen based on the physical limitations of the robot (linkage collision occurs at angles less than θ = 45o) and previous work regarding knee springs with a ratio R = 1/1. Reproduced with permission from [9] c 2017 IEEE.

run with an additional knee up posture. More importantly, this posture (though much less stable than the other linkage configurations) has significant overlap with the configurations at smaller values of θ. From this, it can be deduced that linkage configurations within the range of θ = 45◦ to

21 θ = 105◦ all have regions that overlap, allowing near seamless transition between postures during stable running. Transition speciﬁcally to this knee above hip posture provides additional travel to perform explosive jumps during steady-state running.

5.1.2 Experimental Validation of 5-bar SLIP

The running trends described previously (Sec. 5.1.1) were observed experimentally on the FSU

MiniBoom (Fig. 1.1b). The R = 1/1 linkage ratio was initially constructed at λ1 = λ2 = 15cm. Due to the nature of the motor requirements (explained previously as being represented by the torsional stiﬀness requirements of Fig. 5.1), the robot was unable to perform with this particular mechanism. A secondary set was with λ1 = λ2 = 10cm which produced successful, steady-state running with limited success that was very sensitive to initial conditions. The R = 1/2 ratio

(with linkages λ1 = 10cm and λ2 = 20cm), however, was run at the three different postures from simulation studies (θ = 45◦, θ = 75◦, and θ = 105◦). Each of these configurations were then run at three non-dimensionalized relative leg stiffness values found to produce stable running, ˜ k10% = [8, 12.5, 16.5] (defined in Table 5.1 and restricted by motor and power limitations) over a prescribed range of touchdown angles with attempts to first locate the steepest angle of attack and then moving towards shallower angles until failure to produce stable running. Experimentally, stable running is determined through repeatability and the ability of the system to complete over 22m of lateral travel under the given parameters.

Table 5.1: Single-leg Running Experimental Results of Stability Analysis (this table is reproduced from [9] c 2017 IEEE)

Nominal Spring Lower Upper Range Mean Hip Value Stiﬀness Bound Bound ◦ ◦ ◦ ˜ ◦ ◦ (∆α0, ) (α0, ) (θ0, ) (k10%,-) (α0, ) (α0, ) 16.5 66 70 4 68 45 12.5 63.5 69 5.5 66.25 (Ex. Knee down) 8 60 68 8 64 16.5 67 71.5 4.5 69.25 75 12.5 65.5 70 4.5 67.75 (Knee down) 8 62.5 67 4.5 64.75 16.5 69 70 1 69.5 105 12.5 67.5 69 1.5 68.25 (Knee up) 8 64 65.5 1.5 64.75

22 The obtained data (Table 5.1) demonstrates comparable trends to those obtained through simulation studies. The width of the basin of stability (fifth column of Table 5.1) is generally much greater for the exaggerated knee down posture as opposed to the knee up posture. Additionally, the mean angle follows a similar curve to simulation studies in which a higher nondimensional stiffness usually produces more stable gaits at a more upright posture. Through the spread of these mean angle values between different configurations of posture, the overlap of stable touchdown angles is evident, providing further support for the assumption that on-the-run transitions should be possible between postures while maintaining stable locomotion.

5.2 Jumping

To investigate the eﬀect of linkage ratio manipulation on hopping ability, provided a range of output torques, the previously described simulation model was ﬁrst analyzed to determine the maximum achievable hop height of the 5-bar leg given a motor model based on the U8 T-motor. The leg was extended from minimum length of ζ = 10cm to the maximum possible length of

ζmax = 30cm for R = 1/2 and from ζ = 0cm to ζmax = 30cm for R = 1/1. The actuator input was manipulated to express output percentages of 50%, 75%, and 100% to evaluate the inevitable lower regions of actuator capability during steady-state running. The linkage ratio between the primary and secondary linkage and the motor effort were manipulated for maximum stroke length, singular hopping in simulation via Matlab (Fig. 5.2). The results demonstrated that the hop height is greatke length (∆ζ) is maximized. From this simulation study, it is demonstrated that maximum hop heights are produced for R = 0.54 (50% effort), R = 0.60 (75% effort), and R = 0.62 (1est for ratios between R = 0.54 (50% effort) and R = 0.62 (100% effort) for this particular motor model and leg length. This means that the linkage for Minitaur (R = 1/2) is very close to the optimal ratio for energetic, single hopping to maximize height. This analysis was verified through experimentation utilizing the linkages created for the FSU MiniBoom. Figure 5.3 depicts the results of performing single, one dimensional jumping with linkage ratios of R = 1/1 and R = 1/2. The boom was set to output the prescribed efforts through the full motion of the linkage to maximize hop height. Five separate jumps were performed for each parameter set to measure system consistency. Additionally, the described simulation for single

23 (a) Hopping Deﬁnitions (b)

Figure 5.2: (a) The 1D hopping model performs a single jump, starting near full compression, extending until liftoff is achieved through force balancing between the actuator output and gravity. (b) For a range of different ratios at three designated control efforts (% voltage to motors), the hop height displacement demonstrated optimal ratios of 0.54, 0.60, and 0.62 (for 50%, 75%, and 100% effort, respectively). This figure is reproduced with permission from [9] c 2017 IEEE.

jumping (Sec. 3.2.2) was modiﬁed to include a calculated damping value for the system in order to better capture experimental results,

τ dθ bζ˙ ζ¨ = −g + − (5.1) m dζ m

y¨ = −g + by˙ (5.2)

This linear damping value (b) is meant to encompass sources of error such as boom inertia, joint viscosity, and motor/power supply ineﬃciencies. This value was obtained through running the fsolve Matlab function which use a LevenbergMarquardt algorithm for the simulation at 75% eﬀort for both ratios. Running simulation to manipulate this constant ±15% provides an additional boundary to attempt to capture error in this measurement.

24 (a) (b) (c)

Figure 5.3: A single, powerful jump with the FSU MiniBoom platform begins with the leg at full compression (a) with output torque to both motors until full extension (b). The results of the hop height measured from maximum height of the boom arm are normalized for comparison between R = 1/1 and R = 1/2 (c). Reproduced with permission from [9] c 2017 IEEE.

5.3 Running and Jumping

Establishing a basis of stable running and maximized jumping, it is clear that the next step involves an analysis of the combination of these behaviors. The existent overlap between running with knee down and knee up postures demonstrates the capability of transitioning into a near complete compression posture during steady-state running from a more stable knee down configuration for the ratio R = 1/2 legs. Additionally, since this leg was determined to be almost the optimal ratio for producing maximum jump with the given motor model, the composite controller should be able to overcome significant obstacles during steady state running. Based on the experimental trials performed previously, parameters were determined to maximize system velocity, stability, and obstacle clearance in jumping. For the running portion, the ratio ◦ ◦ R = 1/2 linkage with a knee down configuration (of θ = 45 ), angle of attack of α0 = 68 , and nondimensional stiffness of 12.5 produced a fast, stable gait which was able to hold a steady-state

25 velocity of ≈2.6m/s. To maximize hopping, the leg length is set to 11cm in preparation for obstacle clearance and the output torque is set to 100% (though due to the nature of running, some energy of the motors will be utilized to redirect energy from impact and maintain leg angle during stance). With the R = 1/1 ratio, it was much more diﬃcult to produce stable gaits but a scaled linkage of

λ1 = λ2 = 10cm was fabricated to avoid saturation limitations. With this leg, a single stable gait ◦ ◦ was produced with θ0 = 22.5 , angle of attack α0 = 64 , and nondimensional stiﬀness of 16.5 which reached a speed of ≈2.0m/s. For jump preparation, the leg length was reduced to 7.33cm (which is the geometric scaling of the 11cm used for R = 1/2) and the eﬀort was increased from 85% to 87.5% (avoiding the saturation limitation).

Figure 5.4: Experimental data depicting the run-to-jump results for the R = 1/2 and R = 1/1 legs. Reproduced with permission from [9] c 2017 IEEE.

The results of these experimental trials are depicted in Figure 5.4. The leg approaches the obstacle after it is has established steady-state running then prepares for the jump in a single step by reconﬁguring its leg length. Once it detects ground collision and rotation of the hip beyond the

26 point of contact, it extends with the prescribed output torque, producing an explosive jump over the obstacle (Fig. 5.5). The obstacle in question constructed through stacking 1cm thick plywood and 2cm thick slats of medium density ﬁber board (MDF). The R = 1/1 leg was incapable of clearing even a single perturbation of 1cm while the R = 1/2 leg cleared a 21cm high stack of MDF and plywood, continued running on top of the constructed obstacle, and even managed to stick the landing on the other side with continued stable running on the ground.

Figure 5.5: Photographic timelapse demonstrating the conﬁguration of the R = 1/2 compressing and subsequently clearing the obstacle with its jumping capability. Reproduced with permission from [9] c 2017 IEEE.

27 CHAPTER 6

A PHYSICAL MANIFESTATION OF DYNAMIC SIMILARITY AND SCALING

The ability to directly scale controllers between similar legged systems of varying sizes is a significant contribution to the field of bio-inspired robotics. The complexity of legged robots often involves a vast array of possible control solutions that require significant time commitment to determine and optimize ideal parameters appropriately. Having studied the dynamics of multi- legged gait development and energetic obstacle negotiation, dynamic scaling provides a means of directly translating these strategies to larger or smaller systems (with consideration of actuation limitations). Previously, the process of dynamic scaling has been demonstrated for dynamic legged locomotion through the use of dimensional analysis [32]. Conversion of these principles to a physical system requires additional considerations however, due to the non-conservative nature of reality. To tackle these limitations, a plan of action is set with the following objectives:

1. Design a large scale platform - First, utilizing existing kinematic leg designs and knowl- edge of the desired scale, materials are selected to allow the intended actions of the system.

2. Characterize system limitations - Second, scale the system limitations, such as motor power and loading integrity, in order to isolate sources of failure during the process.

3. Scale limitations - Third, using dynamic scaling laws, apply these limitations to a smaller scale platform to ensure characteristic behavior matching will be dynamically similar.

4. Controller Development - Fourth, design and optimize a controller at the smaller scale for dynamic motion, producing the desired behavior with limitation.

5. Scaling up - Finally, apply the principles of dynamic scaling to the scale up the controller to the larger system, recreating the behavior in a predictable manner.

With these steps in mind, the diﬃcult task of dynamic scaling was approached through use of the ARL Hopper (Fig. 3.2a) and UPenn Thumper (Fig. 3.2b) utilizing the described AER controller previously described.

28 6.1 Designing at the Larger Scale

Since the ARL Hopper already existed, it was decided that a larger scale system would be created in its image to analyze dynamic scaling principles. Researchers were recruited from the University of Pennsylvania’s Kod*lab to approach this task and constructed the UPenn Thumper. Wei-hsi Chen of the Kod*lab led this eﬀort through the conception and development of the UPenn Thumper (Fig. 6.1). He developed this robot with same kinematic design as Minitaur at a 1.5 times larger geometric scale. To accommodate for the increased torque requirement of this system, Thumper uses the T-motor U10 Plus (as opposed to the U8 used by the ARL Hopper and Minitaur). The robotic leg’s mass comes out to 2.762kg which is only a little more than twice the mass of the ARL Hopper.

Figure 6.1: Sketch design for Thumper platform produced by Wei-hsi Chen of the Kod*lab at University of Pennsylvania.

6.2 Characterizing Limitations: System Identiﬁcation

Before analysis was to begin between the two systems, an initial study was done to analyze dynamic scaling on the ARL Hopper to determine the necessary characteristics that should be addressed when scaling between the UPenn Thumper and ARL Hopper. First, a set of steady- state, 1D hopping experiments were performed on the ARL Hopper using the AER controller (Sec.

29 Figure 6.2: Analysis of 1D hopping on the ARL Hopper with variation in controller damping for two distinct system masses.

3.1.2) and varying an input torsional damping via the derivative gain of the motors. Then, to change the systems scale, mass a standardized 0.1kg mass was added to the legs center of mass (COM). The control input was scaled accordingly and the same sweep of controlled damping was performed to evaluate the signiﬁcance of system damping. In Figure 6.5 it was determined that the lack of overlap between the 1.3kg and the scaled 1.2kg systems emphasizes the possible lack of damping that could exist between the ARL Hopper and UPenn Thumper. To accommodate for this observed energetic disparity in this experiment, a system identiﬁcation is performed. To do this, a factor of damping is incorporated into the equations of motion for the 5-bar leg design (Eq. 3.4 and Eq. 3.5),

κ dθ c dθ ζ¨ = ζψ˙2 − gcos(ψ) + (θ(ζ ) − θ(ζ)) − θ˙ (6.1) m dζ 0 m dζ

2ζ˙ψ˙ g cψ˙ ψ¨ = − + sin(ψ) − (6.2) ζ ζ mζ2

30 where c represents a torsional damping constant meant to capture the friction, damping, and other non-conservative entities present in the system and θ is solved to determine the inner linkage half angle from the leg length

ζ2 + λ2 − λ2 θ(ζ) = cos−1 1 2 (6.3) 2λ1ζ Additionally, the value of gravity (g) is manipulated in this analysis to accommodate for the inertial eﬀect of the boom. This was done through derivation of the equations of motion accounting for the mass and construction of the boom as demonstrated previously [33]. With the addition of a counter weight to this system, Equation 19 in [33] becomes

ML + ml/2 − mclc z¨ = g0 2 2 2 l (6.4) ML + ml /3 − mclc where the counter mass (mc = 1kg) and leg/fixture mass (M = 1.439kg) are represented as point masses that are given distances from the center of rotation, (lc = 0.089m and L = 1.372m, respectively) and the boom arm (m = 0.240kg, lb = 1.22m) is represented as a cylindrical rod rotating about its end. With these values, the assumed gravity becomes 9.70m/s2. The damping constant is solved through sweeping different control parameters for a 1D drop test on the ARL Hopper. This is done by first commanding the leg to maintain a centered angle (accounting for the toe extension) then dropping the leg from a varying height between 1 − 5cm. Torsional stiffness (κ, controlled by the proportional motor gain) and damping values (c, controlled by the derivative motor gain) are swept and focus is placed on the dynamics of linear compression (Eq. 6.1). After 5 consecutive drops are obtained, the resulting parameters of damped frequency, initial inflection points, and steady-state rest length are determined as depicted in Figure 6.3b. The Matlab fsolve function is used to determine the torsional damping that most accurately satisfies all 5 drops through starting at the initial inflection point and matching the damped frequency and following two inflection points (labeled in Fig. 6.3b). Using this method and the experimental results obtained from the parameter sweep, calibration curves were developed for system stiffness and damping of the ARL Boom (Fig. 6.4). Through this it was determined that the system mechanical damping of the ARL Hopper is 0.0914Nms/rad and that this increases by a factor of 7.96 with increasing derivative gain through control.

31 (a) Multiple Leg Drops (b) Single Leg Drop Matching

Figure 6.3: Multiple leg drops are performed on the ARL Boom (a) then individual drop events are isolated (b). The necessary output values are extracted including the ﬁrst two minimum peaks, the maximum peak, the period of oscillation, and the steady-state leg length.

(a) ARL Stiﬀness Calibration (b) ARL Damping Calibration

Figure 6.4: Calibration curves for stiﬀness (a) and damping (b) determined using the described fsolve method for the ARL Hopper system.

32 Figure 6.5: A Comparison of the ARL Hopper with added increments of +0.1kg and +0.2kg demonstrates similar steady-state hop heights when scaling is applied to the one dimensional AER controller.

This process was then applied in conjunction with the AER-based, 1D hopping controller used previously (Fig. 6.5) for a single parameter set. In this instance, the ARL Hopper is tested with a controller, then additional mass is added with the application of scaling to the control parameters. Consideration for scaling of the energetic output is taken into account through scaled stall torque of the motor. Since voltage directly relates to this value, the voltage is scaled by the scaling factor for torque to provide the proper limitation on energetic input. The linkage lengths remain unchanged throughout this study, maintaining that the length scaling factor, αl = 1. This means that the system should achieve the same steady-state hop height for each scale of mass. Without scaling, the system is incapable of reaching the same hop heights and achieves a lower steady-state value with added mass as is demonstrated in Figure 6.5. Upon successfully creating consistent system matching between the ARL Hopper with and without added mass, it is time to move toward the larger scale system, UPenn Thumper. The same process of system identiﬁcation utilizing Matlab fsolve was performed on the UPenn Thumper

33 (a) UPenn Stiﬀness Calibration (b) UPenn Damping Calibration

Figure 6.6: Calibration curves for stiﬀness (a) and damping (b) determined using the described fsolve method for UPenn Thumper.

through a series of drop tests. The results are shown in Figure 6.6 and demonstrate that this robot has a torsional damping value of 0.3786Nms/rad. To compare the system parameters between the UPenn Thumper and ARL Hopper, it is necessary to determine the scaling factors that determine their dynamic similarity. In this case, the scaling factors for length (αl) and mass (αm) are already known. The boom inertia of the UPenn

Thumper changed the eﬀective gravity experienced by the leg (gT humper = 10.21) and in order to minimize the disparity between the systems, the counter weight was removed from the ARL Hopper

(gHopper = 10.12) to make αg ≈ 1. Through these three factors, the process of dynamic scaling was applied as outlined in Table 6.1.

Table 6.1: Scaling Parameters Relating the UPenn Thumper and ARL Hopper

Parameter Scale Factor Equation Calculation Scaling Down Scaling Up Length (m) αl l/L 0.30/0.45 0.6667 1.5 2 Gravity (m/s ) αg g/G 10.12/10.21 ≈ 1 ≈ 1 Force (N) αF mg/MG 1.3/2.762 0.4707 2.1245 p p Time (s) αt l/L 0.30/0.45 0.8165 1.2247 −1 Frequency (Hz) αf αt 1/1.2247 1.2247 0.8165 Torque (Nm) ατ αF αl 0.4707(0.6667) 0.3138 3.1867 Damping (Nms/rad) αc ατ αt 0.3138(0.8165) 0.2562 3.9029

34 In order to further ensure dynamic matching between the two systems, the principles of scaling are applied to the motors of the two systems. In this case, the UPenn Thumper utilizes the U10 Plus T-motor [2] while the ARL Hopper utilizes a less costly variant of the T-motor U8 brushless

DC motor, the DYS BE8108-16 100Kv. Using the velocity constant (Kv), motor resistance (Rm), and applied voltage (V ), the stall torque and no-load speed can be estimated by

60 Kt = (6.5) 2πKv

Kv τstall = V (6.6) Rm

ωNL = V/Ke where Ke ≈ Kt (6.7)

From Equation 6.6, the stall torque of the U10 Plus is estimated to be 18.85Nm while the stall torque of the DYS BE8108-16 is 5.9Nm with no-load speeds (calculated from Equation 6.7) of 126 and 157 rad/s, respectively. Scaling the torque from the U10 Plus down to the ARL Hopper via the torque scaling factor (found in Table 6.1) demonstrates this variant U8 motor is actually an ideal candidate for matching dynamic scaling characteristics since the required torque is calculated to be 5.92Nm. The no-load speed, however, should additionally be manipulated to match the scaled down value of 103 rad/s (per the frequency scaling factor in Table 6.1, since angle is agnostic of scale). This is done in code by checking the physical value of velocity of each motor and limiting the torque output to the corresponding value on the speed torque curve for the scaled down U10 Plus model. The resulting speed/torque curves for the described motors are depicted in Figure 6.7. Figure 6.8 demonstrates an experimental analysis of AER controlled 1D, steady-state hopping. Control parameters for AER were swept including the torsional stiﬀness (P-gain of 0.375 and 0.5), driving frequency (values of 2, 3, and 4Hz), amplitude boundary of inner linkage angle modulation (10% and 15%), and driving oﬀset of the sinusoid (0.9π/2 and π/2). Though all 24 combinations were analyzed, only 16 were able to achieve height. Of the 8 that failed, they consistently failed between the two systems providing an initial support to the success of scaling. Of the remaining 16 that achieved steady-state hopping, the compressed leg length, resulting hop frequency, and apex hope height were analyzed for a window of 20-25 consistent hops. Throughout this comparison,

35 Figure 6.7: Theoretical motor speed/torque curves for the T-motor U10 Plus, DYS BE8108-16, and scaled down version of the U10 Plus.

there is minimal error which is most likely due to the abstraction of non-conservative entities through a singular constant when there most likely exists nonlinear effects (i.e. variability of effective gravity through the boom arm angle, system friction and damping, etc). With positive results from 1D hopping, a controller was developed for sagittal plane running. The gait, produced using AER, incorporates angle modulation between two set angles; a touchdown angle is held until impact with the ground is detected then while length actuation is occurring, a liftoff angle is set with a given gain to rotate the hip and provide forward body translation. A preliminary run was performed on the UPenn Thumper to confirm the stability and consistency of the system through two consecutive runs of distances of 21m. The control was then scaled to the ARL Hopper and the same experimental run was performed. The results of these experiments are depicted in Figure 6.9 with measurements of given output parameters for the ARL Hopper and the scaled equivalent values for the UPenn Thumper. This precursory study provides a basis of support for the physical application of dynamic scaling and similarity to legged robotic systems.

36 Figure 6.8: Experimental analysis of a range of diﬀerent control parameters for AER control scaled steady-state hopping between the UPenn Thumper and ARL Hopper.

Figure 6.9: Scaled matching of two dimensional, steady-state running between the UPenn Thumper and ARL Hopper comparing apex of ﬂight phase, minimum compressed leg length, frequency of running, forward system velocity, touchdown angle, and stance angle swept.

37 CHAPTER 7

CONCLUSIONS

The development of control strategies and mechanical design for dynamic legged robots through simulation and prototyping provide the necessary building blocks toward the future of these systems. In the quest to provide platforms that can perform unique capabilities at the optimum level, time and money become very precious commodities. Through dynamic similarity and scaling, insights of mechanical limitations and controller development can be performed on cheaper, small scaled systems and the results scaled to the desired size. In order to approach this problem, an understanding of design and control of these systems must first be established. This is done through streamlined development of gaits, dynamical analysis of running and jumping behaviors, and ultimately the physical manifestation of dynamic similarity and scaling laws. Gait development on a quadrupedal platform is an involved and cumbersome process. That being said, a streamline method for getting initial crawling gaits working on a unique robotic system was implemented using only the motor space for control. A parameter sweep resulted in the development of a gait that was able to walk at 0.52m/s. Through more involved controller design and physical modification to the system to make it more robust, an improved walking gait was implemented achieving a maximum velocity of 0.92m/s. Slight modification of this gait was then used for transitioning to a trot gait, achieving an average speed of ≈ 1.33m/s and a maximum speed of 1.42m/s before the robot left the tracking area. The quest to understand legged dynamics shifted to the two-dimensional realm at this point through an analysis of sagittal plane dynamics via SLIP modeling. The resultant analysis of running dynamics for the 5-bar leg demonstrated that greater stability could be achieved with a more upright posture though there is significant overlap of the basins of stability with changing posture. With the ratio of this particular leg, highly energetic jumping is capable of nearly 2x the robots maximum leg length due to the properties of its kinematic design in conjunction with the particular ratio utilized by Minitaur. This combined with the stable running of the leg provides the capability to perform fast, stable running toward an obstacle, transition to a crouched posture,

38 and subsequently overcome the obstacle through maximizing torque output. The FSU MiniBoom was able to utilize this control strategy to overcome an obstacle over 2/3 of its maximum leg length during a steady-state run of 2.6m/s and return to stable running thereafter, purely based on the dynamics of the system (since the touchdown angle and torque output were fixed values). Once the system dynamics of legged locomotion are understood to a comfortable extend, dynamic similarity and scaling of these systems was explored. Initial analysis demonstrated the ne- cessity to understand the dynamic parameters defining key nonconservative entities such as friction and damping. A system identification method was developed to solve for a damping value meant to encompass the inertia, friction, and damping intrinsic to the mechanical system. Preliminary tests on the ARL Hopper successfully demonstrated the implementation of dynamic scaling laws with mass variation on a single platform. Further work was then done on analyzing the relation between two systems that differ in leg length and actuator capability through dynamic similarity and scaling theory. Through limiting the motors to act similar in code, successful matching of steady-state hopping was performed in one dimension with a precursory attempt at two dimensional running.

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43 BIOGRAPHICAL SKETCH

Daniel Blackman received a Bachelor’s of Science in Biochemistry and a Bachelor’s of Arts in Physics from Edinboro University of Pennsylvania (Edinboro, PA) on May 18, 2013. He then attended Cornell University (Ithaca, NY) to pursue and earn his Master’s of Engineering degree in Biomedical Engineering on May 25, 2014 where he worked as a graduate research assistant under Dr. Christopher Hernandez on bacterial proﬁling via micro-ﬂuidic measurements of cell turgidity. He then began working in the STRIDe Lab at Florida State University (Tallahassee, FL) under the direction of Dr. Jonathan Clarke where his work has focused on the analysis of gait development and leg design towards dynamic, legged locomotion.