key: cord-0469863-ds6le8tw
authors: He, Juntao
title: Modeling, simulation, and optimization of a monopod hopping on yielding terrain
date: 2021-10-28
journal: nan
DOI: nan
sha: 79b6dac72118e4496ebfde9440347c71016ed9b2
doc_id: 469863
cord_uid: ds6le8tw

Legged locomotion on deformable terrain is a challenging and open robo-physics problem since the uncertainty in terrain dynamics introduced by ground deformation complicates the dynamical modelling and control methods. Moreover, learning how (e.g. what controls and mechanisms) to move efficiently and stably on soft ground is a bigger issue. This work seeks to control a 1D monopod hopper to jump to desired height. To achieve this goal, I first set up and validate a discrete element method (DEM) based soft ground simulation environment of a spherical granular material. With this simulation environment, I generate resistive force theory (RFT) based models of the ground reaction force. Then I use the RFT model to develop a feedforward force control for this robot. In the DEM simulation, I use feedback control to compensate for variations in the ground reaction force from the RFT model predictions. With the feedback control, the robot tracks the desired trajectories well and reaches the desired height after five hops. It reduces the apex position errors a lot more than a pure feedforward control. I also change the area of the robots square foot from 1cm^2 to 49cm^2. The feedback controller is able to deal with the ground reaction force fluctuations even when the foot dimensions are on the order of a grain diameter.

Mobile robots have great potential to improve search and rescue, disaster response, factory inspection and package delivery. Legged locomotion has significant advantages over wheeled locomotion on yielding surfaces, such as soil, sand, snow and gravel. However, researchers have mainly focussed on generating stable walking and running gaits on rigid flat [1] or uneven terrain [2] . There are far fewer studies about legged locomotion on yielding ground.

Researchers already made robots to accomplish some locomotion tasks on soft ground. Earlier work used fixed kinematics (fixed gaits) to move on soft ground [3] [4] . Only recently has the community started to look at feedforward control [5] . In [5] , Hubicki et al. present a model and fast optimization formulation which generates accurate motion plans on granular media with relatively short solving times. Reference [6] achieved flat-footed bipedal walking on deformable granular terrain with feedback control, but they did so by seeking to avoid ground penetration.

Reference [7] suggests an active damping controller to reduce the energy cost for monopedal vertical hopping on soft ground, while reference [8] seeks to minimize energy loss on soft-landing using feedforward force control as well as optimal impedance.

In this work, I use feedforward control to develop a periodic gait on soft ground using a ground reaction force model. I test this approach by performing discrete element method (DEM) simulations of the monoped robot hopping on granular materials. To stabilize the target gait, I introduce a feedback term.

To test the force control approach I develop for periodic hopping on soft ground, experimental validation using a real robot on real ground would be preferable. 3. The performance of the robot and its control on different soft substrates can be investigated by simply changing the parameters related to the physical properties of the granular materials.

The structure of the thesis is as follows:

Chapter 2 validates the reliability and accuracy of the Chrono simulation environment by compar- Refrence [3] finds that the granular resistive stresses are linear with depth, but only for low intrusion speeds (less than 0.5 m/s), only in the bulk, i.e., away from the container sides (to avoid Janssen effects [9] ) and at depths both large enough to avoid nonlinearities due to free particles at the surface and small enough to avoid nonlinearities due to compression against the container bottom. For a rigid plate intruder moving in granular media [3] , the lithostatic stresses can be modeled as Considering both reducing force fluctuations and simulation time, 3.2 mm is a good choice for the granular particle radius. Table 2 .1 lists the simulation parameters, for more details, refer to the JSON file in my GitHub repository 1 . The friction is chosen the same as used in [3] . Damping coefficients are not mentioned in [3] , I set them as reasonable values for glass.

To reproduce the stress parameters results for 3 mm glass spheres from the experiments shown in 

As [3] suggests, the stresses in the horizontal and vertical directions are proportional to the penetration depth as Figure 2 .2 shows. By linearly fitting the stress versus depth curves in MATLAB, the stress gradients (α x and α z ) can be calculated as the slopes of those fitted curves. 

where i is x or z, α i,DEM represents the horizontal (x) or vertical (z) direction stress per cubic centimeter of DEM results while α i,Exp represents experimental results of Li et al. [3] . Theᾱ i,Exp term in the denominator represents the mean value of all the experimental results .

As the color map in Figure Table 2 .2: Vertically-constrained (β = 0, γ = π/2) penetration results for three different velocities. Variation of α x and α z in three repeated trials is less than 0.003 N/cm 3 in all cases.

those simulations, the simulation parameters are the same as I discussed above and the rigid plate moves downward in the vertical direction with a constant speed. The orientation of the plate was also fixed (β=0). Figure 2 .2 plots σ x and σ z versus penetration depth z for three constant intrusion velocities V . Lines are fitted over a depth range of 2 cm ≤ z ≤ 7.5 cm. 

Modelling the response of yielding terrains to foot contact is challenging. The resistive forces from the ground are dependent on the granular compaction [11] [12], the intruder kinematics (penetration depth and speed) [3] [13] and the intruder morphology. Reference [13] proposes that the resistive forces consist of a hydrodynamic-like term and a hydrostatic-like term, and investigates rapid intrusion by objects that change shape (self-deform) through passive and active means. [3] presents a granular resistive force theory to predict the forces on objects intruding relatively slowly(where inertial effects are negligible) with different directions and orientations.

As discussed in the second paragraph of Chapter 2, [3] proposes that the granular resistive stresses are linear with depth. To obtain their stress results, Li et al. [3] intruded a rigid thin plate into a granular bed with different plate orientaions and movement directions. And then, they got stress gradients α z,x (β, γ) (see Eq.2.1) by linearly fitting stress results at three different depth (2.54 cm, 5.08 cm, and 7.62 cm). Finally, they performed a discrete Fourier transform on α z,x (β, γ) results over −π/2 < β < π/2 and −π < γ < π to obtain a fitting function. In this chapter, I do something similar but use more modes to get a more accurate representation. The fitting results in this Chapter will be used for further locomoting hopper simulations of our group.

The simulation setup in this chapter is identical to in Figure 2 .1, which shows the planar view of the plate penetrating into the granular domain. The rigid plate is 5 cm × 5 cm × 0.5 cm (area = 25 cm 2 ) and the plate mass is 0.25 kg. More details about the simulation parameters can be found in In Chapter 2, the granular particle container is filled by particles layer by layer from bottom to top, and no further action is taken for the bed preparation. A slightly different filling procedure is used 

To get analytic expressions for α x,z in Eq.2.1, I did hundreds of DEM simulations with different attack angles β and intrusion angles γ which vary from -90 degrees to 90 degrees. The constant intrusion speed is chosen as 3 cm/s. For each β − γ pair, I did three repeat trials, and took the average of α x,z , as in Chapter 2, and then I took absolute values of those α x,z (β, γ) data. Figure 3.3 plots |α x,z (β, γ)|. In Figure 3 values of the raw data to reduce Fourier fitting errors. By performing discrete Fourier transform on the processed raw α z,x data over −π/2 < β < π/2 and −π < γ < π, I can obtain a fitting function.

To balance the accuracy and conciseness of the fitting formulas, I set the Fourier transform order as 2. The fitting functions are:

where A m,n , B m,n , C m,n and D m,n are coefficients of the fitting function and can be found in Table   3 .2. The coefficients for 1st order and 3rd order Fourier fitting can be found in this Google drive file 2 .

To determine the accuracy of the fitting formulas, I generated error maps as shown in fitted α z (Fourier fitted α x ). By comparing the error maps of three Fourier fitting orders, one can see that a second order fit ensures the fitting formula accuracy while limiting the number of fitting parameters. As Figure 3 .5 shows, errors in most regions are close to zero (less than 0.02) and only errors of α z near γ = ±π are near 0.1 N/cm 3 . At the boundaries of the β − γ plane, the physical meaning is the plate switches between extraction and intrusion. It is more difficult to intrude than withdraw, so the values near boundaries see a rapid change along γ. However, low order Fourier fits are less accurate if the raw data changes rapidly. This is why the fit is less accurate near boundaries. Figure 3 .7 shows that the mean and standard deviation of the absolute errors for order 2 is less than 0.0211 N/cm 3 , so the 2nd-order Fourier series approximation can be used as a fast-to-evaluate model of granular resistive stresses, at least for low intrusion velocities and away from the effects of boundaries.

Many applications for mobile robots involve environments that are dangerous or unsuitable for humans. Moreover, due to the inaccessible/unsafe nature of these environments, level rigid ground is rare, so wheeled/treaded robots are often unsuitable due to limited mobility. Legged robots would seem the ideal choice, except that the bulk of research and development of legged robots has focused on hard-ground applications, which are of little use in these challenging environments. In many cases, the terrain is not rigid but, rather, is some kind of deformable granular substrate (e.g., sand, soil, or snow). While there has been recent progress on legged robotic locomotion on these kinds of substrates (e.g., [3] , [13] , [5] , [6] ), a major remaining challenge with hopping/walking/running on these substrates is that they do not return to their pre-impact state.

This represents a permanent energy loss as well as a challenge to maintaining stable gaits. In addition to its relevance to legged locomotion on deformable terrain, this unidirectional ground response represents an interesting variation on the classic "bouncing ball problem" in hybrid dynamics [14] , [4] . While there are many tools for generating and analyzing cyclic trajectories in hybrid dynamical systems (e.g., Poincare analysis, Lyapunov analysis), I bring the tools of trajectory optimization [5] and discrete-element-method (DEM) simulation to bear on the problem.

Specifically, I formulate the search for vertically-constrained hopping gaits on deformable terrain as a trajectory optimization problem with boundary constraints. I solve this problem numerically, obtaining a feedforward control signal. Then, in DEM simulation, I apply this feedforward control signal along with a stabilizing time-invariant feedback law.

In this chapter, I develop a control strategy for a vertically-constrained monopod robot hopping on non-cohesive frictional granular media. The robot is released from the apex of its trajectory and then impacts the bed with velocities ranging from 0 to -4 m/s. When the foot contacts the granular bed, the ground reaction forces are modeled by RFT. Then the equations of motion of the robot has 

Eq. 2.1 shows that the horizontal and vertical resistive forces increase linearly with the intrusion depth. For a 1D hopping monopod robot, the granular substrate can be modeled as a unidirectional Foot dimension 5 cm x 5 cm x 10 cm Foot mass 0.25 kg Body dimension 5 cm x 5 cm x 5 cm Body mass 1.25 kg Stroke limit 0.5 m Table 4 .1: Physical properties of the monopod robot.

spring. The ground resistive force f g is expressed as 

To focus on foot-ground interaction rather than whole-body control, I study a simple monopedal robot hopping vertically as illustrated in 

The hybrid dynamics of the hopping robot are divided into three phases: flight, yielding stance, and static stance. 

3)

where g is the gravitational acceleration (9.81 m/s 2 ). For the following section on robot control, initial conditions are,

where V 0 is the impact velocity.

My first task is to find period one hopping gaits for the vertically-constrained monopod robot. Gait generation is the formulation and selection of a sequence of coordinated leg and body motions that propel a legged robot along a desired path [15] . Specifically, my robot starts impact at a certain initial position and moves vertically downward with an initial impact velocity. The desired motor control periodically returns the robot to the initial impact position with the same CoM velocity magnitude but in the upward direction on each successive hop.

There are various methods of determining the control signal u(t) that results in periodic hopping trajectories (e.g., Poincare analysis [16] , Lyapunov analysis), but I use trajectory optimization [17] to find controls and trajectories that satisfy periodicity constraints. 

To determine the open loop control signal for the robot, I divide the problem into two parts, stance phase and flight phase. The stance-phase optimal control problem can be transformed into a constrained nonlinear program. For the flight phase, no ground forces are applied to the foot, the velocity of body and foot is expected to be equal, and the distance between the body and foot should return to the initial value, so analytic methods are used to determine the corresponding control signal.

Hubicki et al. [5] describe a fast optimal motion planning algorithm for a 1D hopping robot. By formulating an optimal control problem as a constrained nonlinear program (NLP), I utilize a welldeveloped NLP solver (i.e., MATLAB fmincon) to find out the control signal. Control signals u(t)

are chosen as fourth order polynomials because the system has 4 constraints, which determine 4 coefficients of the polynomials. The remaining coefficient is determined by optimizing the cost function. The expression is as follows, For a longer time like 1s can also make the optimization problem converge but it needs more time to solve. In addition, too short a stance time will not be able to develop enough momentum for the robot to jump back to apex given stroke constraints and lack of rate dependence in GRF model.

I choose control effort squared as the cost function:

Reference [8] chooses intrusion depth of the foot as the cost function. I choose the control effort squared because it makes the cost function convex which is good for the convergence of the optimization problem. The resulting formation for the stance-phase nonlinear program is as follows: By setting the stance-phase duration to 0.3 s and boundary conditions as given in Table 4 

After testing the open loop control forces in Chrono simulation, I found that the trajectories of the robot deviate from the expected ones because the ground stiffness approximation is not sufficiently accurate. Therefore, I introduce a feedback controller to stabilize control in stance phase. So, the control strategy is feedforward plus feedback control now. 

The ultimate goal of my work is to design gaits for monopod hopping on deformable ground. For the single hop in the sections above, the robot jumps on undisturbed ground. To challenge my feedforward + feedback controller, I make the monopod to have five hops on disturbed ground. As velocity of the body and foot equal, and make the body-foot distance equal to the initial value. This again shows the robustness of my controller. In addition to the plots in this chapter, I also made corresponding videos of the robot hopping in Chrono. The first video 1 shows five hops on This work proves controlling a monopod to desired height periodically when hopping on yielding terrain is possible. However, our solution for the control force is not optimal, at least in terms of the energy lost to ground. My feature plan is to seek the optimal control that ensures the periodically hopping while minimizes the intrusion depth. Besides, all the experimental results I got are based on DEM simulation. It's interesting to carry out experiment validation on a real robot.

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