I Introduction
In contrast to many popular contemporary legged robots [8, 23, 22], the Penn Jerboa (fig. 0(a)
) is a dramatically underactuated biped: it has 12 DoF (degrees of freedom) and only 4 direct drive actuators
[13, 14, 24]. Moreover, unlike many bipeds, it features small point toes rather than flat feet [5]. Jerboa’s consequent high power density (43.2 w/kg [24]) and its unusual recourse to a high powered (two DoF) tail at the expense of affording only one actuator at each hip of its passive spring loaded legs provokes the question of whether and how its largely dynamical and comparatively more energetic regime of operation can offer performance competitive with that of more conventional legged designs.Because its extreme underactuation precludes recourse to popular control approaches that assume the availability of arbitrary ground reaction forces subject to linear frictional constraints [31, 11, 7, 25, 9], prior work on Jerboa has built on the long tradition of anchoring compositions of dynamical templates [16, 12, 28, 27]. This paper introduces a new entry to the catalogue of spring loaded inverted pendulum (SLIP) [34] template controllers, offering a novel, modelrelaxed strategy for controlling the speed and height of Jerboa using its hip motors. This leaves the tail free to control the pitch and roll (future work). Compared to previous work on Jerboa, this control strategy allows the robot to hop faster with just as much height — albeit potentially with the need for greater traction since the original tailenergized hopping mode [36] explicitly drives energy down the leg shaft, increasing the normal component of the ground reaction forces.
We find fixed points and prove stability for an approximate model of closed loop hopping generated by this control strategy by using assumptions to construct an analytical return map. The fixed points of this simplified system effectively approximate the numerical fixed points of the unsimplified system allowing for an intuitive understanding of how the parameters affect the steady state operating regime of the robot. We test this control strategy in hardware showing a wide range of steady state operating regimes.
Ia Background Literature
Since Raibert, the SLIP model has been used to study legged locomotion[33, 35, 1, 32, 30]. Raibert controlled SLIPlike robots with a fixed thrust in stance to energize hopping and used the touchdown angle to control speed.
Due to the simple nature of the flight map and resets, much of the analytical work on SLIP has focused on approximating the stance map [18, 2]. The fixed points of the analytical return map serve as a map from the many dimensional parameter space to the operating conditions of the robot [26].
Inspired by the work in [18], Ankarali et al. develops an analytical approximation of the stance map for SLIP with damping and gravity by linearizing gravity, yielding a conservation of angular momentum [2]
. Ankarali et al. then approximates the effect of gravity by iteratively estimating the average angular momentum
[6].They extend the results to the stance map for hip energized SLIP by taking the open loop hip torque into account when calculating the average angular momentum[3, 4]. They use inverse dynamics to find the hip torque to get to the target energy and solve an optimization problem for the touchdown angle. Ankarali et al. are unable to analytically find fixed points of the return map and they never implemented the control strategy on a physical robot.
IB Contributions and Organization
With the goal of developing a simple, analytically tractable, control strategy for a hip energized SLIPlike robot, this paper makes three contributions: (i) a simple and novel control strategy for hip energized SLIP where energetic losses from damping are counteracted by choosing a touchdown angle that puts most of the speed at touchdown in the radial direction (eq. 3) and the leg angle is energized using the hip torque (eq. 4); (ii) a closed form analytical approximation of the fixed points for a simplified model of SLIP under the new control strategy (eq. 8); (iii) an extensive empirical study of the implementation of the novel control strategy on the physical Jerboa robot [36].
This control strategy (i) allows for intuitive control of the foreaft velocity(figure 3) and some control over the apex height (figure 3). The analytical predictions (ii) match the fixed points of the numerical return map(table III) and predict with reasonable accuracy the height and speed of the robot (fig 5, fig 5, table IV). The robot (iii) demonstrated stable locomotion with speeds ranging from 0.4 m/s to 2.5 m/s (2 leg lengths/s to 12.5 leg lengths/s) and heights ranging from 0.21 m to 0.27 m (fig 5, fig 5, fig 6).
Section II develops the approximations that yield a closed form return map. Section III introduces further simplifying assumptions affording simple closed form approximation of the fixed points of the return map and compares the accuracy of both relative to their counterparts arising from numerical integration of the original physical model (sec. IIIB). Section IV demonstrates the performance of the controller on Jerboa. Section V wraps up the paper with a discussion of the key ideas and some future steps.
Symbol  Brief Description  Ref 

leg angle  fig. 0(b)  
leg length  fig. 0(b)  
mass  fig. 0(b)  
spring constant  fig. 0(b)  
hip torque  fig. 0(b)  
damping coefficient  fig. 0(b)  
spring rest length  fig. 0(b)  
foreaft position  fig. 0(b)  
height relative to ground  fig. 0(b)  
the state in stance =  sec. IIA  
acceleration due to gravity  eq. 1  
angle of attack gain  eq. 3  
angle of attack  eq. 3  
target angular momentum in stance  eq. 4  
the state at apex =  eq. 5 
Ii Return Map
The ballistic modes (ascent and descent) of the SLIP model are completely integrable; hence the effort in constructing the return map whose fixed points are of interest lies in approximating its nonintegrable stance mode map [19].
Iia Stance Map, Ascent Map, Descent Map, Resets
The nonintegrability [21] of SLIP dynamics
(1) 
introduces the burden of imposing simplifying assumptions yielding physically effective approximations for designers seeking closed form expression and stability guarantees for steady state gaits.
In order to get integrable dynamics we start with two assumptions inspired by the work in [6].
Assumption 1
Since we control to a target angular momentum, , the angular momentum is constant and equal to the target angular momentum.
Assumption 2
The leg angle is small, thus gravity acts radially.
As in [6], we take a Taylor series approximation of the and terms in the post assumption dynamics (eq. 9) centered at , where , yielding
(2) 
a 1 DoF linear time invariant damped harmonic oscillator in feeding forward to excite a first order linear time invariant leg angle integrator in . For the closed form solutions see (eq. 12, eq. 13, eq. 16).
Integrating (2) and following [3] to obtain , the liftoff time, yields the stance map approximation, , , which we display in closed form below in eq. 6 after introducing a number of further simplifying assumptions.
IiA1 Ascent, Descent, and Reset Maps
Due to the single point mass, SLIP follows a ballistic trajectory in flight. Apex is defined by , and touchdown occurs when the toe contacts the ground.
IiB Angular and Radial Control Policies
Early on in our experiments, we found that Raibert stepping [30] did not work well for hip energized hopping because the Coriolis term did not provide sufficient coupling for the hip torque to counteract the radial damping.
Angle of attack control (hereafter, AoA) counteracts the energetic losses from damping by choosing a touchdown angle that puts most of kinetic energy at touchdown into the component; as a result is small. The hip torque then reenergizes directly, without relying on coupling.
Let be the touchdown angle under AoA where is the gain on the touchdown angle (nominally 1) and is the angle of attack. . Since varies with , this yields the constraint equation
(3) 
where is the vertical energy. An approximation to the implicit function for satisfying constraint 3 is presented in (eq. 22).
The AoA gain, , is nominally between and . corresponds to having some initial angular velocity at touchdown. A corresponds to having angular velocity in the opposite direction of travel at touchdown. For this paper we will restrict .
The angular control policy is a PID + feed forward loop which control the leg to a target angular momentum.
(4) 
IiC Constructing the Return Map
Iii Fixed Points and Stability of the Analytical Return Map
Notwithstanding the closed form expression for represented by (5), developing an intuitively useful closed form expression for its fixed points is facilitated by the following simplifying assumptions.
Assumption 3
Across all fixed points, the nondimensionalized time of stance, is constant.
Assumption 4
. ^{1}^{1}1Based on the radial velocity at liftoff, the spring constant, and the damping, this is accurate within 2mm.
Using these assumptions the stance map, derived in Sec. IIA takes takes the form
(6) 
Assumption 5
and (see figure 6 for evidence that this assumption is valid ).
Assumption 6
The change in gravitational potential energy between touchdown and liftoff is negligible compared to the kinetic energy
Assumption 7
where .
See (app. BB) for a derivation of this assumption. Total energy and foreaft velocity are conserved in flight, thus at the fixed point we impose the constraints, and yielding quadratic polynomials in the touchdown state variables (eq. 27, eq. 28). The constraint on energy yields a quadratic polynomial in whose coefficients, are given by elementary functions of the physical parameters and control inputs listed in (eq. 29). In turn, the constraint on foreaft velocity yields a quadratic polynomial in whose coefficients, are given by similar functions that now also depend upon the value of (eq. 30eq. 32). Thus, the fixed points of (eq. 5) are given as
Given the fixed point in touchdown coordinates, we map it backwards in time to get the apex coordinate fixed points:
(8) 
We check stability of the fixed points by evaluating the Jacobian of the return map and checking if its spectral radius is less than 1 at the fixed point.
Iiia Plots of Analytical Fixed Points
Table II has the parameters used in the analytical return map. Figure 3 and 3 plot the apex foreaft speed and height as a function of the control inputs, and . The apex speed shows a significantly decoupling between and ; heavily effects the speed while has almost no effect on the speed. By using (corresponding to speed when ), the resulting relationship between the target and actual speed would be monotonic and zero at zero target speed.
(kg)  

(m)  0.2 
(N/m)  4000 
(Ns/m)  20 
On the other hand, the apex height shows some coupling between and . has a monotonic relationship with height, though increasing increases . This should be thought of as for the higher , has a larger affordance on height. The simple formulation of this control strategy, the decoupling in the control of foreaft velocity, and the angle of attack gain’s consistent effect on height makes this control strategy ideal for hip energized SLIP.
IiiB Accuracy of the Fixed Points
In order to validate our numerous assumption and evaluate the accuracy of the analytical return map, we compared its fixed points (eq. 8) to the fixed points of the numerical return map.
Table III is the error between the fixed points of the analytical return map and the fixed points of the numerical return map over a rectangular approximation of the robot’s operating regime, , . The error is mostly small over the operating regime except for the error in when and are large. These control inputs result in a large , thus violating assumptions 2, 5, and 6.
RMS Error  Percent RMS Error  

0.488 m/s  20.3%  
0.037 m  13.3% 
Iv Experiments
In order to test angle of attack control, we implemented the controller on the boommounted, pitchlocked, Jerboa robot with its tail removed [36].
The control strategy works very well in hardware. The robot achieved speeds of up to 2.5 m/s (limited by kinematics) and heights of up to 0.27 m. The main failure modes were premature touchdown and failing to liftoff. The attached supplemental video has clips of Jerboa hopping at representative steady state set points across the operating regime. Fig. 5 and Fig. 5 contrast the analytical prediction of apex coordinate fixed point with empirical data taken at steady state.
Iva Experimental Setup
The Jerboa robot weighs 3.3 kg, has a max hip torque of 7 Nm from 2 TMotor U8 [38], a 4000 N/m spring leg, and an operating voltage of 16.8 V from an offboard LiPo. Jerboa’s processor is a PWM mainboard from Ghost Robotics[20] which runs angle of attack control at 1 kHz.
IvA1 Implementation Detail
Angle attack control requires an estimate of the robot’s velocity at liftoff (3). We estimated this value using a coarse derivative of the robot’s position estimated with the leg kinematics (touchdown to liftoff for foreaft velocity, and bottom to liftoff for vertical velocity).
Additionally, the motors controllers on Jerboa did not have current/torque control, only voltage control. Fortunately we were still able to use a controller of the same form as eq. 4, though the constant on the gravity compensation term had to be manually tuned.
IvB Empirical Fixed Points
We tested the robot with and . Figure 5 and 5 plot the apex coordinate fixed points for the control inputs that resulted in stable locomotion. Jerboa was able to hop at speeds ranging from 0.4 m/s (2 leg lengths/s) to 2.5 m/s (12.5 leg length/s). As was predicted by the analytical return map, the foreaft speed is controlled by . Additionally, the robot was able to hop at heights ranging from 0.21 m to 0.27 m. As with the analytical return map, the height is controlled by the and where increasing increases the height. Table IV presents the error of the fixed points of the analytical return map compared to the empirical results.
RMS Error  Percent  Max Error  Max Percent  
RMS Error  Error  
0.595 m/s  36.7%  1.328 m/s  82.0%  
0.0264 m  11.6%  0.060 m  26.5% 
The analytical return map does an excellent job predicting the empirical fixed points. Not only are the trends preserved, but the actual values are fairly similar. The height is accurate to about 1/10 of a leg length and the error in speed is concentrated at the higher angular momentums.
IvC Steady State Trajectories
Figure 6 shows the steady state trajectory of the robot with moderate speed and height. The angular momentum asymptotically converges to the target angular momentum over the course of stance without saturating the motors. The resulting leg angle trajectories are very asymmetric. while . This means that the hip torque is always increasing the normal component of the ground reaction forces, rather than decreasing it. Even though , is opposite the direction of travel. This is likely caused by the impact at touchdown and the 4 bar linkage in the leg.
These plots also shows the validity of some of our earlier assumptions. For the lower , the leg angle stays small. Although we assumed angular momentum is constant, it changes quickly over stance, always reaching or slightly overshooting .
Figure 7 is the limit cycle from the empirical trials for and a step response of . This plot shows that the system to robust to changes in input and that increasing increases the energy in the radial subsystem.
V Conclusion
This paper presents a novel hip energized control strategy for a pitch constrained Jerboa. The control strategy allows Jerboa to hop at speeds of up 2.5 m/s. By constructing and validating an analytical return map we found a closed form expression for the fixed points giving insight into how the parameters affect the operating regime of the robot.
Va Discussion
VA1 Energizing the Radial Component Through Resets
With hip energized SLIP, the energetic losses enter through the radial dynamics while the energization happens in the leg angle dynamics. Since the only coupling in stance occurs through the Coriolis terms of the dynamics (which are typically very weak in regimes of physical interest), we energize the radial direction through a smart choice of reset map. By controlling around a touchdown angle that maximizes (eq. 3), the radial direction can be energized at the cost of . The leg angle is then reenergized using a hip torque. One side effect of this strategy is that the asymmetry in the leg angle greatly reduces the traction concerns that comes with hip energized hopping [15].
VA2 High Speed Hopping
VA3 Comparison to Previous Jerboa Controller
Compared to the previous tail energized Jerboa controllers [13, 15, 36, 14], angle of attack control allows Jerboa to hop faster with just as much height. One downside of controlling a robot with angle of attack control is that it can’t hop in place; thus when transitioning from forward hopping to backwards hopping the robot needs to transition from angle of attack control, to tail energized, and back. This maneuver points to: (i) situations where it makes sense to use one strategy vs. the other, and (ii) the need to develop ways of transitioning from tail energized hopping to angle of attack control.
VA4 Model Relaxed Control
VB Future Work
Future work should focus on developing controllers for Jerboa that allow it to use angle of attack control without the constraints on pitch and roll. The previously developed roll controller [40] is a good place to start.
Acknowledgments
This work was supported in part by the US Army Research Office under grant W911NF1710229. We thank J. Diego Caporale for his support with the experimental setup, WeiHsi Chen for his support with scaling, and Charity Payne, Diedra Krieger, and the rest of the GRASP Lab staff for keeping our lab safe and open during these tumultuous times.
References

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Appendix A Return Map Appendices
Aa Maps
AB Stance Dynamics
(9) 
AC Stance Mode ODE Solution
AC1 Radial ODE Solution
The fully simplified radial dynamics (2) are
(10) 
In order to solve equation 10 in a more compact form let , let , and let . With these variables, equation 10 can be written as
(11) 
Assuming , the solution to equation 11 is of the familiar form of a forced spring mass damper.
Where and and are determined by the touchdown states, and
We further simplify the radial flow with and giving
(12) 
Differentiation yields the radial velocity
(13) 
Where .
AC2 Leg Angle ODE Solution
With the analytical approximation for the radial dynamics, we can solve for the leg angle solution. Since angular momentum is conserved, . As with the radial dynamics, approximating with a Taylor series about yields a closed form solution.
(14) 
Using equation 14 along with the radial solution yields
(15) 
AD Time of Liftoff Equation
Liftoff is defined as the time when the force in the leg, . As in [3] we assume the compression time is roughly equal to the decompression time. Thus .
We find , the bottom time, by setting equation 13 equal to zero and solving.
Solving for in with the solutions to the dynamics substituted yields
Where . In our operating regime of interest it is safe to take , , and the .
AE Ascent and Descent Map Derivations
AE1 Flight Trajectory
In flight, the robot follows a purely ballistic trajectory about the center of mass. We describe this trajectory using cartesian coordinate.
(17) 
AE2 Ascent Map
Apex is defined as the state where , thus . From here the ascent map, is derived from equations 17 evaluated at .
AE3 Descent Map
Touchdown occurs when the toe comes in contact with the ground. This is described by the equation . Plugging in the flight solution gives
Whose solution is
(18) 
From here the descent map, is derived from equations 17 evaluated at .
AF Reset Map Derivation
Let be the reset map that maps from the stance state to the flight state at liftoff.
(19) 
Where, and is the liftoff state in stance coordinates.
Let be the reset map that maps from the stance state to the flight state. This map changes coordinates from cartesian to polar.
(20) 
Where .
AG Approximate Solution of Angle of Attack
Starting from equation 3 we use the Taylor series expansion of and the small angle approximation of . We approximate the angle of attack as
(21) 
Equation 21 is quadratic in , with coefficients
Appendix B Details on Angle of Attack Fixed Points
This appendix should be used in conjunction with section III.
Ba Constants in Simplified Stance Map
The constants in equation 6 are defined as
(23)  
(24)  
(25)  
(26) 
BB Derivation of Assumption 7
. Assuming is nominally , then where . From trigonometry, we get that for a given , and where .
BC Solving For The Fixed Points
After assumption 7 the constraints on energy and speed are
(27)  
(28) 
The constrain on energy is quadratic in with coefficients
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