Motor Selection for a Tail

Adding an inertial tail to RHex could enable a variety of new behaviors, however here the focus will be on reorientation in free-fall. The tail created for RHex [92] is an appropriately scaled approxi- mation to that of Tailbot [32], both comprising an approximate point-mass made of brass at about

Figure 2.2: XRL [65] with a new tail, and with approximately sized image of Tailbot [32, 108].This figure originally appeared in [92] is included here courtesy of Tom Libby and Ryan Knopf.

1/10th body mass, attached to a carbon fiber tube of about one body length, as seen in Figure 2.2. While Tailbot was a special-built machine, the tail for RHex must be added to an existing platform as a modular payload [65], and as such the range of motion is significantly lower than Tailbot’s, especially given a safety margin to avoid collision with the body. To compensate, the RHex tail design targets a slightly higher effectivenessε=1.29 (this is the ratio of tail to body rotation [92]) so as to afford the same 90◦body correction capability as Tailbot.

To mitigate the integration task, both Tailbot’s and XRLs tail actuators were for simplicity cho- sen to be the same as their wheel/leg motors. But to maximize the performance of the tail, a more careful study is warranted. In the spirit of [44], we can define a single dynamical task and a number of performance metrics, and then calculate the optimal performance of all available motors [124]. In that paper, the optimal gear ratio was calculated via numerical optimization on one performance metric — here we find an analytical solution1. The power,P, is given for each motor and we now

1_{Anecdotally, having this analytical solution to the gear ratio reduced the simulation time for all 1,546 motors con-} sidered from 171s to 0.175s.

seek to determine the minimal completion time,t, (parametrized by morphology) for a rotationθ0

as a function of power rather than the inverse (to find the minimum power needed to complete a rotation in a fixed time). The optimal no load speed, ωm, (a proxy for gear ratio) and resulting

completion time,t, functions are (see [93] for full derivation),

ωm= kωθ0P Ib 1+1 ε 2!1/3 ; t= ktθ02Ib P 1+1 ε 1/3 . (1)

whereIb is the body inertia, and the constantskω ≈3.156,kt ≈1.547. Other metrics to consider

are physical (size, mass), electrical (current and voltage available), and thermal. The thermal cost of a tail for inertial self-righting is in general small due to the very small time scales, however some motors may still overheat (see below for more detailed thermal modeling). Now, following [44], whose numerical optimization step does not require the restriction to the linear dynamics used to derive (1) and which can incorporate these additional metrics, the performance of all commercial motors [124] can be compared. Out of the 1,546 motors considered, 82 of them met the length (<30mm), weight (<200g), and minimum completion time (<0.5s) constraints. Of those, the chosen motor was the third fastest, only 22% slower than what would be the optimal motor. The optimal gear ratio for our motor would be 27:1, the 28:1 gear ratio used is the closest commercially available2.

As an empirical validation of design (including the motor selection), we conducted a series of initial tests (Figure 2.3) to see how large a body rotation can be achieved by a relative tail rotation of about 155◦, which is limited by geometry. The robot was dropped nose first from a height of 1.36m (over 8 times the standing height and 2.7 times the body length). The body angle was measured from an IMU and regulated to horizontal by a simple PD controller. The motor was able to rotate the tail, and the robot, in 0.35 seconds to within 5◦ of level, or just more than one body length of fall, and maintain that with no more than 4◦ overshoot. This test used the entire 155◦ range of relative tail motion, rotating the body a maximum of 89.7◦_{before hitting the hard stop. From these}

two final positions we can calculate an averageεn=1.38, which matches our approximate estimate 2_{This analysis could be made more accurate by considering a current limit and more complicated controller.}

Figure 2.3: XRL self-righting in a fall.

Figure 2.4: XRL surviving a run off a cliff outdoors.

ofε=1.29 to within 7% and is reasonable considering the errors involved in measuring inertia [148] and manufacturing. As a comparison, the robot was also dropped with no tail activation, causing the front two legs to snap as well as some minor internal damage. Thus if the robot tasks requires a fall from this height, it is definitely survivable assuming it successfully reorients to land within about 5◦of level3_.

To demonstrate this new ability for XRL in a practical task, the second set of experiments was conducted outdoors, running along and then away from a 62 cm (3.8 times the hip height or 1.2 body-length) cliff. The robot’s inertial sensors detect the cliff upon initial body pitch, then actuate the tail according to the previously described closed loop control policy, and the robot lands on its feet (Figure 2.4). Another test with XRL running from a cliff with apassivetail confirmed that it would land nose first.