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Research Papers

# A Three-Dimensional Printed, Nonassembly, Passive Dynamic Walking Toy: Design and AnalysisPUBLIC ACCESS

[+] Author and Article Information
Christian L. Treviño

Robotics and Motion Laboratory,
Department of Mechanical Engineering,
The University of Texas at San Antonio,
One UTSA Circle,
San Antonio, TX 78249

Joseph D. Galloway, II

Robotics and Motion Laboratory,
Department of Mechanical Engineering,
The University of Texas at San Antonio,
One UTSA Circle,
San Antonio, TX 78249

Pranav A. Bhounsule

Robotics and Motion Laboratory,
Department of Mechanical Engineering,
The University of Texas at San Antonio,
One UTSA Circle,
San Antonio, TX 78249
e-mail: pranav.bhounsule@utsa.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received July 16, 2017; final manuscript received June 6, 2018; published online September 21, 2018. Assoc. Editor: Sarah Bergbreiter.

J. Mechanisms Robotics 10(6), 061009 (Sep 21, 2018) (8 pages) Paper No: JMR-17-1211; doi: 10.1115/1.4040634 History: Received July 16, 2017; Revised June 06, 2018

## Abstract

In this paper, we present the redesign and analysis of a century old walking toy. Historically, the toy is made up of two wooden pieces including a rear leg and a front leg and body (as a single piece) that are attached to each other by means of a pin joint. When the toy is placed on a ramp and given a slight perturbation, it ambles downhill powered only by gravity. Before the toy can walk successfully, it needs careful tuning of its geometry and mass distribution. The traditional technique of manual wood carving offers very limited flexibility to tune the mass distribution and geometry. We have re-engineered the toy to be three-dimensional (3D) printed as a single integrated assembly that includes a pin joint and the two legs. After 3D printing, we have to manually break-off the weakly held support material to allow movement of the pin joint. It took us 6 iterations to progressively tune the leg geometry, mass distribution, and hinge joint tolerances to create our most successful working prototype. The final 3D printed toy needs minimal postprocessing and walks reliably on a 7.87 deg downhill ramp. Next, we created a computer model of the toy to explain its motion and stability. Parameter studies reveal that the toy exhibits stable walking motion for a fairly wide range of mass distributions. Although 3D printing has been used to create nonassembly articulated kinematic mechanisms, this is the first study that shows that it is possible to create dynamics-based nonassembly mechanisms such as walking toys.

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## Introduction

Passive dynamic robots are machines that can walk or run downhill using only their natural dynamics. These robots are made by assembling individual parts and by iteratively tuning their dynamics by placing external weights and testing on a downhill ramp. Each time a new robot is assembled, it needs to go through this iterative tuning. In this paper, we explore the creation of a passive dynamic walking toy using three-dimensional (3D) printing. The approach has multiple advantages: (1) the mass distribution can be easily tuned by modifying the CAD dimensions and/or print properties (e.g., in-fill), (2) the joints can be printed within the 3D printed body to create an articulated toy without the need for assembling or for additional tooling, and (3) once a successful design iteration is achieved, multiple toys can be 3D printed with minimal tuning. The disadvantages are the long print times for simple designs and the difficulty in fine control of the mass distribution because of the assumptions made by cad software (e.g., uniform mass distribution). However, it is conceivable that the disadvantages will be overcome with further advances in 3D printing.

## Background and Related Work

Passive dynamic toys have been in existence for almost 100 years as evidenced by a number of patents [15]). We discuss two toy designs:

1. (1)The patent of the Wilson walkie [5] is shown in Figs. 1(a) and 1(b). The toy has two legs, each of which connects to a body through a hinge joint. When the toy is launched on a downward incline, the side-ways rocking of the body lifts a foot off the ground. The off-ground foot then swings forward to complete a step. The same sequence is repeated with the other foot and enables the toy to descend downhill.
2. (2)The design of Ravert's toy [4] is shown in Fig. 2. The toy consists of a body (number 10) and two legs (number 11). The body is connected to each leg by a hinge joint (number 12). There is a mechanical stop (number 13) to prevent the legs from overextending. There is a groove in the body (number 17) and the ramp has a guide rib (number 16) that prevents the toy from veering sideways. Both toys are able to descend downhill when given a slight perturbation.

The term “passive dynamic walking” was coined by McGeer [6] who first demonstrated that a human-like frame can descend downhill powered only by gravity. While the toy designs mentioned earlier had a rather stiff gait, McGeer's robot had a more human-like appearance. To simplify balance, McGeer constrained his walker to 2D (only front-back or sagittal plane motion) by pairing the inner legs to each other and similarly the outer legs. A true 3D passive dynamic walking robot was built by Collins et al. [7]. Design features included were curved feet with special guide rails, counter swinging arms, and soft heels to improve stability in side-to-side (coronal plane) and left-right (axial plane) directions. A 2D passive dynamic runner was created by Owaki et al. [8] by adding a torsional spring in the hip and linear springs in the legs. All of these robots are statically stable when their legs are splayed. An interesting idea is to create a legged robot that cannot stand stably but can move stably (like a bicycle). A walking toy based on this concept was created by Coleman and Ruina [9], and more recently, a hopping robot was created by Steinkamp [10].

Gomes and Ahlin [11] created an (almost) passive dynamic walking robot, a rimless wheel, that can move on level ground. Their design involves a hip spring that winds up as the rimless wheel moves from midstance to support transfer. The wheel's spoke touches the ground with almost zero speed ensuring a smooth support transfer. On the subsequent step, from support transfer to midstance, the spring unwinds to power the rimless wheel. Other successful level ground walking robots exploiting passive dynamic ideas use some or other form of actuation, such as reactive pendulum [12,13], vibration of beam [14], and impulsive ankle push-off [15]. In such cases, actuation is necessary to overcome the energy losses during locomotion (e.g., collisions and internal dissipation due to friction).

Another class of toys are those which rely only on their kinematics to walk, unlike the dynamics-based toys discussed earlier. The simplest kinematic-based toys have a single input (e.g., a crankshaft) which is connected to multiple outputs (e.g., legs) through a series of gears, levers, and linkages. When the crankshaft is rotated, it sets the gears and/or levers and/or linkages in motion that in turn cause the legs to move, and allow the toy to walk. One example is the Strandbeest, which means “beach beast” when translated, created by Dutch artist Theo Jansen [16]. The original Strandbeest was several feet tall and made up of Polyvinyl Chloride pipes, but has recently been scaled down and 3D printed. In the Strandbeest, the crankshaft is connected to propellers. The wind causes the propellers to rotate. This rotation is then converted to reciprocating motion of the legs through a series of linkages, allowing the Strandbeest to walk. More recently, Coros et al. [17] have created a computational framework that allows the nonexperts to create animated characters which can then be 3D printed. Lipson et al. [18] provide a history of mechanisms from the Cornell Kinematic Mechanisms collection that can be 3D printed. Since these toys rely only on their kinematics, they are easier to tune in comparison to dynamics based toys discussed in this paper.

There has been limited work done on 3D printing dynamically walking toys. The Wilson Walkie has been 3D printed by Haberland [19] and is shown in Fig. 1(c). The body and the two legs are printed separately. A nail connects the two legs to the body and also serves as a hinge joint. The toy is able to descend downhill just like the actual wooden Wilson walker shown inset in Fig. 1(c). More recently, Stöckli et al. [20] have created a toy similar to the Ravert walker [4] and Coleman's tinkertoy [9] but individual pieces were 3D printed and then assembled together. However, it should be possible to change the design to print the entire system monolithically using the same material with a different in-fill for support structure as done in this paper or by using separate support material that dissolves when dipped in appropriate liquid (usually water). The latter is a feature available with newer 3D printers.

Our toy design is loosely based on the Ravert toy walker [4]. In our toy design, the legs are arranged back-to-back instead of sideways (see Fig. 3). The main novelty of this work is the demonstration that dynamic walking toys can be 3D printed as a nonassembly mechanism. More specifically, 3D printing allows us to design and print the toy as a single assembly inclusive of the hinge joint, thus eliminating the manual assembly of the toy. 3D printing has allowed us to tune the mass distribution, while preserving the toy's shape and allowing for quick iteration on the leg geometry. Our final design evolved in just six design iterations and was able to walk down a 7.87 degree incline with minimal post processing and is described in Sec. 3. Further, we created a computer simulation of the model and analyzed the motion and walking stability in Sec. 4. A discussion of the results is presented in Sec. 5, followed by conclusion in Sec. 6.

## Mechanical Design

Figure 3 shows the final toy design. The toy is based on the mascot of the University of Texas at San Antonio (UTSA) named “rowdy” The Roadrunner. The toy has two legs. The front leg is fixed to the body, while the rear leg is attached to the body through a single hinge joint. The hinge joint allows for rotation of the moving leg in the sagittal plane. When the toy is placed on a ramp and perturbed slightly, it walks downhill alternating between the two legs. The toy was 3D printed on Ultimaker 2. A video of the making of the toy and walking motion is available online.2 We discuss specific aspects of the toy next. More details are available under “Supplemental Data” tab for this paper on the ASME Digital Collection.

###### Hinge Design.

A hinge joint is used to connect the rear leg to the body and to allow relative motion between the body and the leg. Figure 4(a) shows a sectional view of the toy and indicates the tolerances on the hinge joint. We designed the hinge joint as follows. We created a circular hole at the top-end of the leg. Next, we aligned the hole with a circular shaft that is designed on the body. The hole was made slightly bigger than the shaft to allow the rear leg to move relative to the shaft on the body. Note that the rear leg is not attached to the body and is floating in free space within the CAD design. While printing the toy, the rear leg is held by support material (polylactic acid (PLA) with lower material fill than used to print the body). Once the printing was finished, we broke off the support material to allow the rear leg to move.

Our initial prototypes of the printed toy had too little clearance between the circular hole and the shaft so that the rear leg could not move freely. This was because of the limited printing resolution of our desktop 3D printer. We had to manually increase the clearance between the hole and the shaft until we produced a functional hinge joint. The hinge joint has a little bit of play in the transverse and coronal plane, and so the toy tends to veer in one direction during walking sometimes.

###### Leg and Feet Design.

There are two legs in our toy design—a front leg that is fixed to the body and a rear leg that is connected to the body using a hinge joint. The rear foot is a circular arc with its center at the hinge joint as shown in Fig. 5. The same radius of curvature is used for the front foot. The shaded area near the rear foot in Fig. 4(a) shows the tolerances around the rear foot. We also found that making the rear (moving) leg to be slightly longer than the front leg leads to a slightly better walking performance than legs of the same length. It is unclear why this asymmetry improves the walking performance.

The toy walks easily on rough surfaces such as wood stock, cardboard, and plywood, yet it slips on aluminum. We created grooves in the transverse direction along the bottom of the feet in attempts of increasing the roughness without successful aid. However, by adding rubber soles to the bottom of the feet, the toy was able to walk on a 3D printed ramp with a slope of 7.87 deg. The rubber soles were cut from heat shrink tubing and were glued firmly to the base of the feet.

###### Tuning the Mass Distribution.

The mass distribution should be such that the center of mass (COM) is on the rear leg but beneath the pin joint when the legs are held against each other. We were able to tune the design in CAD so that the mass distribution was between the two legs. However, when the first prototype was printed, the center of mass was found to be in the front resulting in the toy sitting on its beak. The discrepancy was due to the cad software's estimation of the center of mass based on a uniform mass distribution which is not true when 3D printed.

To tune the mass distribution, we attached known weights to the rear of the toy until it achieved a static standing position. Based on the tuned weight, we then re-adjusted the CAD design by enlarging the feathers and hollowing the beak. The modification was successful in achieving the necessary mass distribution to achieve stable downhill walking.

###### Support Material.

The role of the support material is to act as a scaffold to brace and hold overhanging and undercut structures while printing, so that the printed plastic does not sag under gravity. In our case, the same PLA that was used for printing the toy was used as support material, except with a lower material fill percentage, allowing for the manual removal once the printing was completed.

In our toy design, there are three distinct places where support material is used:

1. (1)Support material is placed between the rear leg and the body as shown by the shaded areas in Fig. 4(a). The support material is manually detached using a screwdriver to allow the rear leg to move.
2. (2)Support material is placed in the hollow section of the beak along the transverse direction to allow 3D printing of the beak as shown in Fig. 4(b). The support material is inside the toy and cannot be removed once printed.
3. (3)A layer of support material is deposited on the 3D printer bed before the toy is printed. The support material is removed after the printing is completed. We noticed that if the toy is directly printed on the bed, then the face which is placed on the bed gets warped due to the heat of the bed. Thus, the presence of the support material on the bed prevents the side adjoining the bed from warping.

###### Three-Dimensional Printing.

Before 3D printing, the CAD design is preprocessed using CURA, an open source slicing software. At this point, we specify 3D printing parameters that include the scaling, the temperature of the nozzle, and the in-fill percentage. We choose an in-fill percentage of 80% for the toy. A higher in-fill percentage increases the print time but too low a value decreases the weight of the toy and this affects its performance. Also, we chose an in-fill percentage of 10% for the hollow section of the beak. Ideally, a zero in-fill percentage leads to a perfectly hollow section but a finite value is needed to act as a supporting structure for the upper layers of the printed toy. The chosen value of 10% is the minimal value that allows enough support as the toy is printed. Both the in-fill percentages were decided by trial and error. A visualization of the pattern used by the 3D printing software is shown in Fig. 4(b). Next, the toy is 3D printed on Ultimaker 2, a desktop hobby-grade 3D printer using a 2.85 mm diameter PLA filament. The print time is about 12 h, which was 3 h less than the time estimated by CURA. After the printing is complete, we manually remove the supporting material that holds the rear leg in place using a screwdriver.

###### Testing.

The testing of the toy was initially done on a wooden ramp, but subsequently the toy was able to walk on a 3D printed PLA ramp. First, we set up a ramp inclination. Then, we placed the toy on the ramp and perturbed it slightly by tilting the toy backward or forward and releasing it. If the toy was not able to descend downhill, we increased or decreased the inclination a little and tried again. If the toy stopped, then the slope was too small and we needed to increase it. However, if the robot slid downhill or overturned, then the slope was too big and we needed to decrease the slope. For our toy design, we have found that a $7.87deg$ or 0.1373 rad incline works best. By adding rubber feet to the soles, the toy is able to descend smooth surfaces such as aluminum.

## Computer Simulation

###### Model.

A caricature of the model with the dimensions is shown in Fig. 5. The robot consists of two pieces: (1) The body and front leg as a single unit shown in light gray color (we designate this piece as the front leg), and (2) the rear leg shown in dark gray color. A pin joint connects the two legs together. To derive equations of motion, we use a line of reference fixed on each leg as follows: (1) The reference line for the front leg is along its rear edge with the origin at the pin joint and (2) the reference line for the rear leg is along its front edge and with the origin at the pin joint. The masses of the front and the rear leg are m1 and m2, respectively, and the location of the COM measured relative to the line of reference is c1 and w1 for the front leg and c2 and w2 for the rear leg as shown in the figure. There is a hard stop that limits the angle between the legs (i.e., the angle between the two reference lines on the legs) to α. The leg length is r, gravity is g, and ramp slope is γ. We assume the following kinematics: the leg on the ground is the stance leg with absolute angle q1 and absolute angular velocity u1, and the other leg is the swing leg with relative angle q2 and relative angular velocity u2Display Formula

(1)
The equations of motion for a single step are derived using the sequence of phases and transition shown in Fig. 6. Equation (1) represents the sequence of motion for the walker. The equation is a graphical method of representing the motion sequence for a hybrid dynamical system. The phases of motion are defined between the arrows and the transition conditions are defined above the arrows. For example, One DOF, Front Leg Stance indicates that the walker moves as a single unit (i.e., the swing leg is locked to the stance leg due to the angle stop). Then, the transition condition, Front-to-rear Support, occurs wherein the support is transferred from the front leg to the rear leg. This transition leads to phase Two DOF, Rear Leg Stance. In this phase, the rear leg is the stance leg and rolls freely, and the front leg is the swing leg that pivots about the rear leg—a two degree-of-freedom system. From Eq. (1), it can be seen that the system is either a 1 DOF or a 2 DOF system during a single step. For the 1 DOF system, we use the angular momentum balance about the contact point of the stance leg to derive an equation for the absolute angular acceleration, $q¨1$. For the 2 DOF system, we use the angular momentum balance about the contact point and the pin joint to derive equations for the absolute angular acceleration $q¨1$ and $q¨2$. The front-to-rear and rear-to-front transitions are smooth and we make an appropriate switch of leg angles (i.e., current stance leg becomes swing leg and vice versa). The front-leg stop and rear-leg stop are non-smooth transitions where the colliding swing leg comes to a stop and the velocity of the stance leg after collision is found by using the conservation of angular momentum about the pin joint. See the supplementary material for additional details (which is available under “Supplemental Data” tab for this paper on the ASME Digital Collection.).

###### Motion Analysis: Periodic Motion and Stability.

Given an initial state for the system $x≡[q1,u1,q2,u2]$ and the simulation parameters (see Table 1), the equations of motion are integrated forward in time to simulate the motion of the system. We are interested in the periodic motion of the toy. Since the complexities of the equations of motion preclude an analytical solution, we resort to numerical techniques.

Periodic motion is found by treating a walking step as a Poincaré map [6]. Given the state of the system at a particular instant (e.g., front-leg stop, rear-to-front support transfer) of the walking step, xn, and the state at the same instant but on the next step, xn+1, one can find a function F that relates the two, xn+1 = F(xn). To find periodic motions, we find a fixed point x* such that x* = F(x*). This is equivalent to finding the zeros of G(x*) = x* − F(x*). This is the same as solving three equations in three unknowns (the Poincaré section reduces the state space from 4 to 3). However, we can reduce the dimension of G from 3 to 1 by choosing the Poincaré section to be at any instant the toy is a single degree-of-freedom system. For example, any of the following stances from Fig. 6 will suffice (a), (b), (c), (f), or (g) (A similar reduction can be done for the kneed passive dynamic walker, see Supplemental results which are available under “Supplemental Data” tab for this paper on the ASME Digital Collection). We use the instant just after the rear leg comes to a stop after colliding with the front leg (see Fig. 6(a)). The initial states at this instant are $xr≡[q1,u1]$ and are used in F and G.

The stability of the walking motion is evaluated from the Jacobian of the function F at the fixed point, i.e., $J(xr*)=(∂F(xr*)/∂xr)$. The system is stable if the largest eigenvalue is less than 1 and unstable otherwise [21]. In our case, there are two eigenvalues, but one eigenvalue is zero and corresponds to the perturbation in the direction of the Poincaré section. Thus, the other non-zero eigenvalue gives the stability of the periodic motion.

We used matlab to numerically evaluate the functions F and subsequently G. To obtain F, we start the simulation at the instant when the front leg is the stance leg and the rear leg is resting against the front leg (see Eqs. (1b) and (1c)). Subsequently, each phase is integrated using ode113 with a tolerance of 10–12, and the appropriate transition conditions are applied in the order given in Eq. (1). Thus, given the initial state at the instant, the toy moves as a single piece, xr, F returns the state at the same instant but at the next step. The function G can be numerically obtained once F is found. Next, we use fsolve with an accuracy set to 10–10 to compute the fixed point. The fixed point of the limit cycle is $xr≡[−0.0194674275,−1.8975774734]$. Then, we use the central difference of 10–5 to evaluate the Jacobian of F and used eig to find the two eigenvalues. As expected, one eigenvalue was zero and the other value was 0.436 (<1) indicating a stable fixed point. Figure 7 shows the phase portrait of the limit cycle.

We compare the fidelity of the model against the toy. We use the open source software Tracker [22] to analyze the motion of the toy. Tracker is able to create time traces of user defined points on the video. We use time traces of the hinge point and points on the two legs to obtain the absolute angles of each leg. Figure 8 shows the comparison between the model and the experimental data for a trial run on the ramp. As seen from the plot, the simulation is able to capture the features of the experiment reasonably well. Time snapshots comparing the animations are in Fig. 9 and in the online video.3

###### Parameter Study.

Figure 10 shows the effect of varying three robot parameters, the center of mass of the body (w1 and c1) and the leg length (r) on the two limit cycle parameters: the cyclic stability as measured by the maximum eigenvalue and the step velocity, which is nondimensionalized by dividing by $gr$. We chose these parameters because they have a substantial effect on the behavior of the toy. The robot parameters were normalized against their nominal values as given in our successful design and presented in Table 1. For a given leg radius, we used the simulation technique described in Sec. 4 to evaluate the maximum eigenvalue and velocity as a function of center of mass parameters w1 and c1. We repeated the calculation for three distinct r values to obtain the plots. Figure 10 (top panels (a), (b), and (c)) corresponds to the maximum eigenvalue, while the bottom panels (d), (e), and (f) correspond to the velocity. Each pair (a)–(d), (b)–(e), and (c)–(f) corresponds to a distinct r value.

The partial white region in (a), (b), (d), and (e) indicates that no limit cycles were found in that region. Note that there are no white regions in (c) and (f) indicating that the range of solutions increases as r increases. This trend is consistent: as r is decreased to zero, the region of feasible solutions decreases and there are no limit cycles when r = 0. From the top three plots, it can be seen that the maximum eigenvalues lie in the range of 0.05–0.6 for a broad range of parameters (note that eigenvalue greater than 1 indicates unstable limit cycle and stability increases as the eigenvalue approaches 0). There is no distinct trend in the eigenvalues and it tends to be discontinuous in some regions. However, we note that the maximum eigenvalues are lowest in (c) on the bottom right corner. This happens when the center of mass along the leg length, c1, is moved downward and the fore-aft offset, w1, is decreased or moved forward relative to our design. The nondimensional velocity is between 0.02 and 0.035 with maximum velocities occurring at the top right side of each plot. That is, when c1 moved further down and fore-aft offset w1 is moved toward the rear. The velocities seem to be smoother and have a more consistent trend than the eigenvalues except in (D) near the white region or where the limit cycles solutions disappear. We found that the limit cycles disappear as w1 is moved to the edge of the rear leg axis, that is w1 ∼ 0. Our design had a maximum eigenvalue of 0.4360 and nondimensional velocity of 0.0281 and indicated by the black dot in Figs. 10(b) and 10(e), respectively.

## Discussion

We have redesigned an antique walking toy such that it can be 3D printed as a single piece with an integrated hinge joint. After minimal postprocessing, the toy is able to walk downhill using its natural dynamics. Then, we created a computer model of the toy, found periodic motion (limit cycles) using Poincaré section, and analyzed motion stability using the eigenvalue of the Jacobian of the limit cycle. We found stable limit cycles for a wide range of parameters. However, all limit cycles had a very slow walking speed, around 0.03 (nondimensionalized by gravity and leg length).

The use of 3D printing allows us to design and print the complete toy, including the hinge and the rear (moving) leg, as a single assembly. While this eliminates the time needed for assembling the toy, it also makes the design nonmodular. Thus, if the hinge joint or the rear leg breaks, then the complete toy needs to be 3D printed again. The use of 3D printing allows us to tune the mass distribution, specifically the location of the center of mass, while preserving the shape of the toy. The mass distribution is critical for the toy to walk on the ramp. However, the CAD's estimation of the center of mass is different from the actual center of mass and depends on 3D printing parameters, like scaling and fill percentage. As such, we had to tune the mass distribution by trial and error. The 3D printing preprocessing software cura does not estimate the center of mass. However, an ability to simulate the 3D printed toy (before actually printing it) would clearly allow for faster tuning. Another issue we faced was that the 3D printer was not able to print the hinge joint to the clearance that we specified in CAD. Ultimately, we had to decrease these tolerances, which lead to play in the hinge joint in both of the transverse and coronal directions. During some of the trials, the play in the transverse direction caused the toy to veer to a particular side.

To enable steady-state walking, the total energy loss at each step should balance the potential energy gained. To find the energy loss, we proceed as follows. We first compute a periodic solution. Next, we compute the total energy of the system, the sum of kinetic and potential, with respect to the current stance leg at the beginning of each phase of motion described by Eq. (1). Then by subtracting energy from adjacent phases, we can compute the energy loss in each phase. The sum of all energy losses gives the net gain in potential energy. We found that there are three sources of energy loss: (1) the collision between body and the rear leg (phase (f) Front-leg Stop Eq. (1)) accounting for 53% of energy loss, (2) the friction between the rear leg and the ground (phase (e) Two dof, rear leg stance Eq. (1)) accounting for 26% of energy loss, and (3) collision of the rear leg with the front leg (phase (j) rear leg stop Eq. (1)) accounting for the remaining 21%. Since the feet are arcs of a circle with the center at the pin joint, the leg length is equal to the radius of the circle. Thus, the support transfer is smooth and without energy losses. This is the most striking difference between the design and most passive walkers as the latter have their primary energy loss during support transfer due to collision [6,7,9,10,23].

The mass distribution is a very important parameter. Since the rear leg is very light compared to the front leg and body, its contribution to the center of mass may be ignored, and a good approximation of the center of mass is that of the front leg and body assembly. The center of mass needs to be below the pin joint but should be located over the rear leg when the toy is placed in a static standing position on level ground with its legs touching each other. If the center of mass is above the front leg or beyond, the toy tends to sit on its beak. If the center of mass is beyond the rear leg, then the toy sits on its feathers. If the center of mass is above the pin joint, then the toy is unable to rock back-and-forth (similar to an inverted pendulum) and cannot walk. Our dynamic analysis (see Fig. 10) indicates that the following design features are desirable for achieving successful walking motion for a given slope: (1) increase the radius (r) of the leg, (2) move the center of mass away from the hinge joint (c1), and (3) move the center of mass offset toward the rear of the toy (w1). Similar observation was made by McGeer on a different model of walking [6].

The largest eigenvalue of the Jacobian of the Poincaré map indicates walking stability. A value less than 1 indicates a stable system and unstable otherwise. Typical passive-dynamic walkers are only mildly stable at best by this measure, with their largest eigenvalues rarely less than about 0.6 in magnitude [24]. However, our analysis indicates that for a range of parameters, the eigenvalues are between 0.05 and 0.6, indicating a highly stable system in comparison to other passive dynamic walkers. However, like other passive dynamic walkers, the toy had a very slow walking speed of about 0.03 (nondimensionalized by $gℓ$).

Our work has several limitations which we list next. The location of the center of mass is key to walking performance. However, getting the mass distribution correct required elaborate trial and error. This was because 3D printing does not create homogenous objects, and thus, the actual center of mass was different from what we designed it to be in the cad software. The print time for the 157 gm toy (dimensions 15 cm × 5 cm × 8 cm) is around 12 h. This is a significant bottleneck and mass production does not seem feasible. We have searched only for a single period one-limit cycle and one cannot rule out other limit cycles, bifurcations, and chaos as observed in other passive walkers [25,26].

## Conclusions

We have re-created an antique walking toy using 3D printing. The use of 3D printing allows us to print an integrated design that includes joints and moving parts as a single assembly, and helps tune the mass distribution and leg geometry quickly to realize a working prototype. The toy is able to walk down a 7.87 deg incline using only its mass distribution and geometry. Computer simulation of the model indicates a stable limit cycle with largest eigenvalue of 0.26.

The results from this paper can be used as a starting point in creating dynamically balanced walking robots using 3D printing. Further extensions can include 3D printing joints and actuators as a single integrated piece that will lead to more practical robots [27].

## Acknowledgements

• NSF grant IIS 1566463 to P. Bhounsule.

• C. Treviño was supported by a Valero Scholarship from the College of Engineering at UTSA.

• J. Galloway was supported by the Office of Undergraduate Research scholarship and the McNair Scholars Program at UTSA.

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Gomes, M. W. , and Ahlin, K. , 2015, “ Quiet (Nearly Collisionless) Robotic Walking,” IEEE International Conference Robotics and Automation (ICRA), Seattle, WA, May. 26–30, pp. 5761–5766.
Iida, F. , Dravid, R. , and Paul, C. , 2002, “ Design and Control of a Pendulum Driven Hopping Robot,” IEEE/RSJ International Conference Intelligent Robots and Systems, Lausanne, Switzerland, Sept. 30–Oct. 4, pp. 2141–2146.
Zoghzoghy, J. , and Hurmuzlu, Y. , 2014, “ Pony II Robot: Inertially Actuated Baton With Double-Action Pendulums,” ASME Paper No. DSCC2014-6189.
Reis, M. , and Iida, F. , 2014, “ An Energy-Efficient Hopping Robot Based on Free Vibration of a Curved Beam,” IEEE/ASME Trans. Mechatronics, 19(1), pp. 300–311.
Collins, S. H. , and Ruina, A. , 2005, “ A Bipedal Walking Robot With Efficient and Human-like Gait,” IEEE International Conference on Robotics and Automation, Barcelona, Spain, Apr. 18–22, pp. 1983–1988.
Jansen, T. , 2016, “ Strandbeest,” World Wide Web Electronic Publication, Delft, The Netherlands, accessed July 11, 2018,
Coros, S. , Thomaszewski, B. , Noris, G. , Sueda, S. , Forberg, M. , Sumner, R. W. , Matusik, W. , and Bickel, B. , 2013, “ Computational Design of Mechanical Characters,” ACM Trans. Graph. (TOG), 32(4), p. 83.
Lipson, H. , Moon, F. C. , Hai, J. , and Paventi, C. , 2005, “ 3D Printing the History of Mechanisms,” ASME J. Mech. Des., 127(5), pp. 1029–1033.
Haberland, M. , 2007, “ Make Your Own Wilson Walkie,” Cornell University, Ithaca, New York, accessed July 11, 2018,
Stöckli, F. , Modica, F. , and Shea, K. , 2016, “ Designing Passive Dynamic Walking Robots for Additive Manufacture,” Rapid Prototyping J., 22(5), pp. 842–847.
Strogatz, S. H. , 2006, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Publishing, New York.
Brown, D. , 2016, Tracker, Video Analysis and Modeling Tool, World Wide Web Electronic Publication, Aptos, CA.
Stiesberg, G. , van Oijen, T. , and Ruina, A. , 2017, “ Steinkamp's Toy Can Hop 100 Times but Can't Stand Up,” ASME J. Mech. Rob., 9(1), p. 011017.
Bhounsule, P. A. , Cortell, J. , Grewal, A. , Hendriksen, B. , Karssen, J. D. , Paul, C. , and Ruina, A. , 2014, “ Low-Bandwidth Reflex-Based Control for Lower Power Walking: 65 Km on a Single Battery Charge,” Int. J. Rob. Res., 33(10), pp. 1305–1321.
Garcia, M. , Chatterjee, A. , Ruina, A. , and Coleman, M. , 1998, “ The Simplest Walking Model: Stability, Complexity, and Scaling,” ASME J. Biomech. Eng., 120(2), pp. 281–288.
Thuilot, B. , Goswami, A. , and Espiau, B. , 1997, “ Bifurcation and Chaos in a Simple Passive Bipedal Gait,” IEEE International Conference on Robotics and Automation, Albuquerque, NM, Apr. 25, pp. 792–798.
MacCurdy, R. , Katzschmann, R. , Kim, Y. , and Rus, D. , 2016, “ Printable Hydraulics: A Method for Fabricating Robots by 3D Co-Printing Solids and Liquids,” IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, May. 16–21, pp. 3878–3885.
View article in PDF format.

## References

Fallis, G. , 1888, “ Walking Toy,” U.S. Patent No. 376588.
Bechstein, B. , 1912, “ Improvements in and Relating to Toys,” UK Patent No. 7453.
Mahan, J. J. , and Moran, J. F. , 1909, “ Toy,” U.S. Patent No. 1007316.
Ravert, W. , 1932, “ Walking Toy,” U.S. Patent 1,860,476.
Wilson, J. E. , 1938, “ Walking Toy,” U.S. Patent No. 2140275.
McGeer, T. , 1990, “ Passive Dynamic Walking,” Int. J. Rob. Res., 9(2), p. 62.
Collins, S. , Wisse, M. , and Ruina, A. , 2001, “ A Three-Dimensional Passive-Dynamic Walking Robot With Two Legs and Knees,” Int. J. Rob. Res., 20(7), p. 607.
Owaki, D. , Koyama, M. , Yamaguchi, S. , Kubo, S. , and Ishiguro, A. , 2011, “ A 2-d Passive-Dynamic-Running Biped With Elastic Elements,” IEEE Trans. Rob., 27(1), pp. 156–162.
Coleman, M. , and Ruina, A. , 1998, “ An Uncontrolled Walking Toy That Cannot Stand Still,” Phys. Rev. Lett., 80(16), pp. 3658–3661.
Steinkamp, P. , 2017, “ A Statically Unstable Passive Hopper: Design Evolution,” ASME J. Mech. Rob., 9(1), p. 011016.
Gomes, M. W. , and Ahlin, K. , 2015, “ Quiet (Nearly Collisionless) Robotic Walking,” IEEE International Conference Robotics and Automation (ICRA), Seattle, WA, May. 26–30, pp. 5761–5766.
Iida, F. , Dravid, R. , and Paul, C. , 2002, “ Design and Control of a Pendulum Driven Hopping Robot,” IEEE/RSJ International Conference Intelligent Robots and Systems, Lausanne, Switzerland, Sept. 30–Oct. 4, pp. 2141–2146.
Zoghzoghy, J. , and Hurmuzlu, Y. , 2014, “ Pony II Robot: Inertially Actuated Baton With Double-Action Pendulums,” ASME Paper No. DSCC2014-6189.
Reis, M. , and Iida, F. , 2014, “ An Energy-Efficient Hopping Robot Based on Free Vibration of a Curved Beam,” IEEE/ASME Trans. Mechatronics, 19(1), pp. 300–311.
Collins, S. H. , and Ruina, A. , 2005, “ A Bipedal Walking Robot With Efficient and Human-like Gait,” IEEE International Conference on Robotics and Automation, Barcelona, Spain, Apr. 18–22, pp. 1983–1988.
Jansen, T. , 2016, “ Strandbeest,” World Wide Web Electronic Publication, Delft, The Netherlands, accessed July 11, 2018,
Coros, S. , Thomaszewski, B. , Noris, G. , Sueda, S. , Forberg, M. , Sumner, R. W. , Matusik, W. , and Bickel, B. , 2013, “ Computational Design of Mechanical Characters,” ACM Trans. Graph. (TOG), 32(4), p. 83.
Lipson, H. , Moon, F. C. , Hai, J. , and Paventi, C. , 2005, “ 3D Printing the History of Mechanisms,” ASME J. Mech. Des., 127(5), pp. 1029–1033.
Haberland, M. , 2007, “ Make Your Own Wilson Walkie,” Cornell University, Ithaca, New York, accessed July 11, 2018,
Stöckli, F. , Modica, F. , and Shea, K. , 2016, “ Designing Passive Dynamic Walking Robots for Additive Manufacture,” Rapid Prototyping J., 22(5), pp. 842–847.
Strogatz, S. H. , 2006, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Publishing, New York.
Brown, D. , 2016, Tracker, Video Analysis and Modeling Tool, World Wide Web Electronic Publication, Aptos, CA.
Stiesberg, G. , van Oijen, T. , and Ruina, A. , 2017, “ Steinkamp's Toy Can Hop 100 Times but Can't Stand Up,” ASME J. Mech. Rob., 9(1), p. 011017.
Bhounsule, P. A. , Cortell, J. , Grewal, A. , Hendriksen, B. , Karssen, J. D. , Paul, C. , and Ruina, A. , 2014, “ Low-Bandwidth Reflex-Based Control for Lower Power Walking: 65 Km on a Single Battery Charge,” Int. J. Rob. Res., 33(10), pp. 1305–1321.
Garcia, M. , Chatterjee, A. , Ruina, A. , and Coleman, M. , 1998, “ The Simplest Walking Model: Stability, Complexity, and Scaling,” ASME J. Biomech. Eng., 120(2), pp. 281–288.
Thuilot, B. , Goswami, A. , and Espiau, B. , 1997, “ Bifurcation and Chaos in a Simple Passive Bipedal Gait,” IEEE International Conference on Robotics and Automation, Albuquerque, NM, Apr. 25, pp. 792–798.
MacCurdy, R. , Katzschmann, R. , Kim, Y. , and Rus, D. , 2016, “ Printable Hydraulics: A Method for Fabricating Robots by 3D Co-Printing Solids and Liquids,” IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, May. 16–21, pp. 3878–3885.

## Figures

Fig. 1

Wilson Walkie [1]: (a) front view, (b) side view, and (c) A 3D printed Wilson Walkie and wooden toy in the inset [2]

Fig. 2

Ravert toy: (a) perspective view, (b) toy duck, and (c) section view of the toy duck (taken from [3])

Fig. 3

The final working prototype

Fig. 4

Sectional view of the design: (a) hinge design with tolerances. All dimensions are in cm. Shaded areas represent hollow sections; (b) in-fill pattern created by CURA, the postprocessing software for the 3D printer.

Fig. 5

Robot model for simulation

Fig. 6

Sequence of phases and transitions used for simulation model: a single and doubled curved arrow on the figure indicates that the toy is a one or two degree-of-freedom system in that particular phase

Fig. 7

Phase portrait for the front and rear leg. See corresponding letters in Fig. 7 and Eq. (1).

Fig. 8

Comparing simulation with experimental data. Positions of various points on the toy as a function of time. (a) Pin joint, (b) bottom corner of the front foot (nearest to the rear foot), and (c) bottom corner of the rear foot (nearest to the front foot).

Fig. 9

A single step of the walker: (top panel) One step from video. (Bottom panel) Animation from the simulation. See the video online for a comparison.3

Fig. 10

Effect of varying r, w1, and c1 on maximum eigenvalue and velocity. (a)–(c) Maximum eigenvalue and (d)– (f) velocity nondimensionalized by gr. Each parameter r, w1, and c1 is nondimensionalized by its nominal parameter given in Table 1. The values for our design are shown using a black dot.

## Tables

Table 1 Simulation parameters

## Errata

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