also of the order of inverse heavy neutrino mass scale, that is why LFV
couplings of leptons with doubly charged Higgs bosons are strongly suppressed,
$\delta_{L,R}-l-l$ off-diagonal lepton couplings are suppressed by
$1/m_{N}^{2}$ when comparing to diagonal cases.

For reasons discussed in Czakon:2002wm  and more extensively in Gluza:2002vs ,
we take seesaw diagonal light-heavy neutrino mixings. It means that $W_{1}$
couples mainly to light neutrinos, while $W_{2}$ couples to the heavy ones.

To summarize, unlike in the HTM case, the $H^{\pm\pm}-l^{\mp}-l^{\mp}$ vertex
does not depend on the light neutrino mixing. With $v_{L}=0$ the MLRSM
realizes the seesaw type-I mechanism and the light neutrino mass is due to the
existence of additional heavy neutrino states and $v_{R}$ scale.

We should note that it is not natural and very hard to create non-decoupling
mixings for non-diagonal $K_{L}$ and $K_{R}$ matrix elements, even when some
symmetries are considered in type-I seesaw models Gluza:2002vs .

###  7.4 Supplemental material for phenomenological studies of $H^{\pm\pm}$
scalar particles

Diagrams in Fig. 4 present the contribution from the singly and doubly charged
scalar particles to lepton flavour violating processes and to muon
$(g-2)_{\mu}$. Those diagrams contain vertices $H^{\pm\pm}-l_{i}-l_{j}$ and
$H^{\pm}-l_{i}-\nu_{j}$ which origin from the Yukawa part of the Lagrangian,
combining the Standard Model Yukawa term with the triplet part Eq. (29)

${\mathscr{L}}_{Y}={\mathscr{L}}_{Y}^{\Phi}+{\mathscr{L}}_{Y}^{\rm\Delta}=-\>{y}_{ij}\>\overline{L^{i}_{L}}\>\Phi\>{l^{j}}_{R}\;+\mathcal{Y}_{ij}\>\overline{{L^{i}_{L}}^{c}}\>i\sigma_{2}\>\Delta\>L^{j}_{L}\;+\;{\rm
h.c.}$ (66)

From Eq. (66) we obtain the interaction between charged leptons and a doubly
charged scalar and the interaction of a singly charged scalar with a charged
lepton and neutrino. Taking into account Eq. (37) and Eq. (28) and keeping in
mind that $y_{ij}\propto\frac{1}{v_{\Phi}}$ is a SM diagonal matrix

$\mathcal{V}^{\pm\pm}=\left\\{\begin{array}[]{lll}l_{i}^{+}-l_{j}^{+}-H^{--}&=&i\>\left(\mathcal{Y}_{ij}+\mathcal{Y}_{ji}\right)\\\
l_{i}^{-}-l_{j}^{-}-H^{++}&=&i\>\left(\mathcal{Y}_{ij}^{*}+\mathcal{Y}_{ji}^{*}\right)\\\
\end{array}\right.,$ (67)

$\mathcal{V}_{\Delta}^{\pm}=\left\\{\begin{array}[]{c}\tilde{\nu}_{i}-l_{j}^{+}-H^{-}=\frac{i}{\sqrt{2}}\>\cos{\beta}\>\left(\mathcal{Y}_{ij}+\mathcal{Y}_{ji}\right)\\\
\nu_{i}-l_{j}^{-}-H^{+}=\frac{i}{\sqrt{2}}\>\cos{\beta}\>\left(\mathcal{Y}_{ij}^{*}+\mathcal{Y}_{ji}^{*}\right)\\\
\end{array}\propto\frac{\sqrt{v_{\Phi}^{2}-2v_{\Delta}^{2}}}{v_{\Phi}\cdot
v_{\Delta}}\right.,$ (68)

$\mathcal{V}_{\Phi}^{\pm}=\left\\{\begin{array}[]{c}\tilde{\nu_{i}}-l_{i}^{-}-H^{+}=i\>\sin{\beta}\>y_{i}\\\
\nu_{i}-l_{i}^{+}-H^{-}=i\>\sin{\beta}\>y_{i}\end{array}\propto\frac{\sqrt{2}v_{\Delta}}{v_{\Phi}\sqrt{v_{\Phi}^{2}-2v_{\Delta}^{2}}}\right..$
(69)

Vertices $\mathcal{V}_{\Delta}^{\pm}$ and $\mathcal{V}_{\Delta}^{\pm\pm}$
comes from the same part of the Lagrangian and they break the lepton flavour.
Vertex $\mathcal{V}_{\Phi}^{\pm}$ is proportional to $v_{\Delta}$ while vertex
$\mathcal{V}_{\Delta}^{\pm}$ is inversely proportional to the triplet VEV and
dominates up to $v_{\Delta}\sim 10^{6}$ eV. Since we are interested in lower
regions of $v_{\Delta}$ values, its effect is negligible. So, with a good
approximation for low values of $v_{\Delta}$

$\mathcal{V}_{\Delta}^{\pm}\equiv\mathcal{V}^{\pm}\simeq\frac{1}{\sqrt{2}}\mathcal{V}^{\pm\pm}.$
(70)

As we discussed in section 3, the branching ratios of the radiative and
$\mu$-to-$e$ conversion depends on the one-loop form factors. From Eq. (9), we
can read them explicitly

$\displaystyle A_{L}(q^{2})$ $\displaystyle=$
$\displaystyle-\frac{(\mathcal{Y}^{*})_{ei}(\mathcal{Y})_{\mu
i}}{24\sqrt{2}G_{F}\pi^{2}}\Big{(}\frac{1}{12M_{H^{\pm}}^{2}}+\frac{f(r,s_{i})}{M_{H^{\pm\pm}}^{2}}\Big{)},$
$\displaystyle A_{R}$ $\displaystyle=$
$\displaystyle-\frac{(\mathcal{Y}^{*}\mathcal{Y})_{e\mu}}{192\sqrt{2}G_{F}\pi^{2}}\Big{(}\frac{1}{8M_{H^{\pm}}^{2}}+\frac{1}{M_{H^{\pm\pm}}^{2}}\Big{)},$
(71)

where $f(r,s_{i})$ is given by:

$\displaystyle f(r,s_{i})$ $\displaystyle=$
$\displaystyle\frac{4s_{i}}{r}+\log(s_{i})+\Big{(}1-\frac{2s_{i}}{r}\Big{)}\sqrt{1+\frac{4s_{i}}{r}}\log\Big{(}\frac{\sqrt{r}+\sqrt{r+4s_{i}}}{\sqrt{r}-\sqrt{r+4s_{i}}}\Big{)},$
(72) $\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{-q^{2}}{m_{H^{\pm\pm}}^{2}}\mbox{,}\quad
s_{i}=\frac{m_{i}^{2}}{m_{H^{\pm\pm}}^{2}}.$ (73)

We have checked with earlier literature and the analytic forms for the CLFV
processes are given as Chakrabortty:2015zpm ; Chun:2003ej ; Kakizaki:2003jk ;
Akeroyd:2009nu ; Dinh:2012bp :

#### Radiative lepton decay $\bm{l_{i}\to l_{j}\gamma}$:

The branching ratios of radiative decay processes can be given by:

$\text{BR}(l_{i}\rightarrow
l_{j}\gamma)=384\pi^{2}(4\pi\alpha_{em})|A_{R}|^{2}~{}\text{BR}(l_{i}\rightarrow
l_{j}\nu_{l_{i}}\overline{\nu}_{l_{j}}).$

Therefore the BRs for various radiative decays can be written as:

$\begin{array}[]{lr}\begin{array}[]{rcl}\text{BR}(\mu\rightarrow
e\gamma)&=&\frac{\alpha_{em}}{192\pi}\frac{|({\mathcal{Y}}^{\dagger}\mathcal{Y})_{e\mu}|^{2}}{G_{F}^{2}}\Big{(}\frac{1}{M_{H^{\pm}}^{2}}+\frac{8}{M_{H^{\pm\pm}}^{2}}\Big{)}^{2}\;\text{BR}(\mu\rightarrow
e\bar{\nu}_{e}\nu_{\mu}),\\\ \text{BR}(\tau\rightarrow
e\gamma)&=&\frac{\alpha_{em}}{192\pi}\frac{|({\mathcal{Y}}^{\dagger}\mathcal{Y})_{e\tau}|^{2}}{G_{F}^{2}}\Big{(}\frac{1}{M_{H^{\pm}}^{2}}+\frac{8}{M_{H^{\pm\pm}}^{2}}\Big{)}^{2}\;\text{BR}(\tau\rightarrow
e\bar{\nu}_{e}\nu_{\tau}),\\\
\text{BR}(\tau\rightarrow\mu\gamma)&=&\frac{\alpha_{em}}{192\pi}\frac{|({\mathcal{Y}}^{\dagger}\mathcal{Y})_{\mu\tau}|^{2}}{G_{F}^{2}}\Big{(}\frac{1}{M_{H^{\pm}}^{2}}+\frac{8}{M_{H^{\pm\pm}}^{2}}\Big{)}^{2}\;\text{BR}(\tau\rightarrow\mu\bar{\nu}_{\mu}\nu_{\tau}),\\\
\end{array}&\begin{array}[]{rcl}\text{BR}(\mu\rightarrow
e\bar{\nu}_{e}\nu_{e})&=&100\%,\\\ \text{BR}(\tau\rightarrow
e\bar{\nu}_{e}\nu_{\tau})&=&17.83\%,\\\
\text{BR}(\tau\rightarrow\mu\bar{\nu}_{\mu}\nu_{\tau})&=&17.41\%.\\\
\end{array}\end{array}$

The contribution of $H^{\pm\pm}$ to the branching ratios is eight times larger
than by $H^{\pm}$ because of the difference in a magnitude of couplings
between $\mathcal{V}^{\pm}$ and $\mathcal{V}^{\pm\pm}$ in Eq. (70), in
addition the amplitude is proportional to the particles charge (which gives an
additional factor of 4).

#### Three body decays $\bm{l\to l_{i}l_{j}l_{k}}$:

$\begin{array}[]{lcrrr}\text{BR}(\mu\rightarrow
eee)&=&\frac{1}{4G_{F}^{2}}\frac{|(\mathcal{Y}^{\dagger})_{ee}(\mathcal{Y})_{\mu
e}|^{2}}{M_{H^{\pm\pm}}^{4}}\;\text{BR}(\mu\rightarrow e\bar{\nu}\nu),&\\\
\text{BR}(\tau\rightarrow
l_{i}l_{j}l_{k})&=&\frac{S}{4G_{F}^{2}}\frac{|(\mathcal{Y}^{\dagger})_{\tau
i}(\mathcal{Y})_{jk}|^{2}}{M_{H^{\pm\pm}}^{4}}\;\text{BR}(\tau\rightarrow\mu\bar{\nu}\nu),&\qquad
S=\left\\{\begin{array}[]{lll}1&\mbox{if}&j=k\\\ 2&\mbox{if}&j\neq k\\\
\end{array}\right..\\\ \end{array}$

####  $\bm{\mu}-$to$-\bm{e}$ conversion

In the computation of conversion rate $\mu$ to $e$, both form factors
contribute and the analytic form can be written as

$\displaystyle\text{CR}(\mu\mathcal{N}\rightarrow e\mathcal{N}^{*})$
$\displaystyle=$
$\displaystyle\frac{\Gamma_{conv}}{\Gamma_{capt}}\cong\frac{2\alpha_{em}^{5}G_{F}^{2}m_{\mu}^{5}Z_{eff}^{4}Z|F(q^{2})|^{2}}{\Gamma_{capt}}\Big{|}8A_{R}+\frac{2}{3}A_{L}\Big{|}^{2}.$
(74)

Therefore, the $\mu$-to-$e$ conversion ratio in the nuclei field can be given
as Dinh:2012bp ; Chakrabortty:2015zpm :

$\displaystyle\text{CR}(\mu\mathcal{N}\rightarrow e\mathcal{N}^{*})$
$\displaystyle=$
$\displaystyle\frac{\Gamma_{conv}}{\Gamma_{capt}}\cong\frac{\alpha_{em}^{5}}{36\pi^{4}}\frac{m_{\mu}^{5}}{\Gamma_{capt}}Z_{eff}^{4}Z|F(q^{2}=-m_{\mu}^{2})|^{2}\times$
$\displaystyle\Bigg{|}\frac{(M_{\nu}^{\dagger}M_{\nu})_{e\mu}}{2v_{\Delta}^{2}}\Big{[}\frac{5}{24m_{H^{\pm}}^{2}}+\frac{1}{m_{H^{\pm\pm}}^{2}}\Big{]}+\frac{1}{2v_{\Delta}^{2}m_{H^{\pm\pm}}^{2}}\sum_{l=e,\mu,\tau}(M_{\nu})^{\dagger}_{el}f(r,s_{i})(M_{\nu})_{l\mu}\Bigg{|}^{2},$

where: |  ${\Gamma_{capt}}\quad-\;\mbox{total muon capture rate (see Tab.~{}\ref{Gcapt})}$,
---|---
|  $Z_{eff}\quad-\;\mbox{effective charge for the muon in the 1s state}$,

| $\Gamma_{capt}\;[\mbox{s}^{-1}]$ | $\Gamma_{capt}\;[\mbox{eV}]$ | $Z_{eff}$
---|---|---|---
${}^{197}_{\phantom{1}79}\mbox{Au}$ | $13.07\times 10^{6}$ | $8.60\times 10^{-9}$ | 33.5
${}^{48}_{22}\mbox{Ti}$ | $2.59\times 10^{6}$ | $1.71\times 10^{-9}$ | 17.5
${}^{27}_{13}\mbox{Al}$ | $0.7054\times 10^{6}$ | $0.4643\times 10^{-9}$ | 11.5

Table 17: Total muon capture rate and effective charge for
${}^{197}\mbox{Au}$, ${}^{48}\mbox{Ti}$ and ${}^{27}\mbox{Al}$ Kitano:2002mt .

.

#### Muon $({g-2})_{\mu}$

In case of doubly charged scalars, cumulative effects of muon $(g-2)_{\mu}$
and lepton flavor violation have been discussed in details in Leveille:1977rc
; Moore:1984eg ; Chakrabortty:2015zpm  and for a triplet scalar it has been
discussed in Fukuyama:2009xk .

Figure 15:  Feynman diagrams represent the contribution to $(g-2)_{\mu}$
within HTM

Contribution to muon $(g-2)$ from doubly and singly charged scalar are shown
in Fig. 15. Final formulas for $H^{\pm\pm}$ and $H^{\pm}$ reads

$\displaystyle\left[\Delta a_{\mu}\right]_{H^{\pm\pm}}$ $\displaystyle=$
$\displaystyle-\sum_{l}f^{l}\times\Bigg{\\{}\frac{2m_{\mu}^{2}|\mathcal{V}_{\mu
l}^{{\pm\pm}}|^{2}}{8\pi^{2}}\underbrace{\int_{0}^{1}dx\Bigg{[}\frac{\Big{\\{}(x^{3}-x^{2})+\frac{m_{l}}{m_{\mu}}(x^{2}-x)\Big{\\}}}{\Big{(}m_{\mu}^{2}x^{2}+(M_{H^{\pm\pm}}^{2}-m_{\mu}^{2})x+(1-x)m_{l}^{2}\Big{)}}\Bigg{]}}_{C_{1}}$
(77) $\displaystyle-\frac{m_{\mu}^{2}|\mathcal{V}_{\mu
l}^{{\pm\pm}}|^{2}}{8\pi^{2}}\underbrace{\int_{0}^{1}dx\Bigg{[}\frac{\Big{\\{}x^{2}-x^{3}+\frac{m_{l}}{m_{\mu}}x^{2}\Big{\\}}}{\Big{(}m_{\mu}^{2}x^{2}+(m_{l}^{2}-m_{\mu}^{2})x+M_{H^{\pm\pm}}^{2}(1-x)\Big{)}}\Bigg{]}\Bigg{\\}}}_{C_{2}},$
$\displaystyle\left[\Delta a_{\mu}\right]_{H^{\pm}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{\nu}\frac{m_{\mu}^{2}|\mathcal{V}_{\mu\nu}^{\pm}|^{2}}{8\pi^{2}}\underbrace{\int_{0}^{1}dx\Bigg{[}\frac{(x^{3}-x^{2})}{\Big{(}m_{\mu}^{2}x^{2}+(M_{H^{\pm}}^{2}-m_{\mu}^{2})x\Big{)}}\Bigg{]}}_{C_{3}}.$
(78)

$f^{l}$ is a symmetric factor equals to 4 for $l=\mu$ and 1 otherwise. The
term proportional to the $C_{1}$ integral is connected with the diagram a) in
Fig. 15, the term with the $C_{2}$ integral corresponds to Fig. 15 b).
Equation (78) presents a contribution from a singly charged particle $H^{\pm}$
(Fig. 15 c) ) to $(g-2)_{\mu}$. Since both $\mathcal{V}^{\pm}$ and
$\mathcal{V}^{\pm\pm}$ vertices are comparable, see Eq. (70), the
contributions from different diagrams depend mostly on $C_{1}$, $C_{2}$ and
$C_{3}$ integrals. Fig. 16 shows that the strongest contribution $(g-2)_{\mu}$
comes from the doubly charged scalar $H^{\pm\pm}$.

Figure 16: $C_{1}$, $C_{2}$ and $C_{3}$ integrals Eqs. (77)-(78) as a function
of charged scalars mass $M_{H^{\pm}}=M_{H^{\pm\pm}}\equiv M_{H}$.

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# Sim-to-Real Learning for Bipedal Locomotion Under
Unsensed Dynamic Loads

Jeremy Dao, Kevin Green, Helei Duan, Alan Fern, Jonathan Hurst *This work is
supported by the NSF Grant No. IIS-1849343, DGE-1314109, and DARPA Contract
W911NF-16-1-0002.All authors are with Collaborative Robotics and Intelligent
Systems Institute, Oregon State University, Corvallis, Oregon, 97331, USA.
Email: {daoje, greenkev, duanh, afern<EMAIL_ADDRESS>
###### Abstract

Recent work on sim-to-real learning for bipedal locomotion has demonstrated
new levels of robustness and agility over a variety of terrains. However, that
work, and most prior bipedal locomotion work, have not considered locomotion
under a variety of external loads that can significantly influence the overall
system dynamics. In many applications, robots will need to maintain robust
locomotion under a wide range of potential dynamic loads, such as pulling a
cart or carrying a large container of sloshing liquid, ideally without
requiring additional load-sensing capabilities. In this work, we explore the
capabilities of reinforcement learning (RL) and sim-to-real transfer for
bipedal locomotion under dynamic loads using only proprioceptive feedback. We
show that prior RL policies trained for unloaded locomotion fail for some
loads and that simply training in the context of loads is enough to result in
successful and improved policies. We also compare training specialized
policies for each load versus a single policy for all considered loads and
analyze how the resulting gaits change to accommodate different loads.
Finally, we demonstrate sim-to-real transfer, which is successful but shows a
wider sim-to-real gap than prior unloaded work, which points to interesting
future research.

##  I Introduction

The primary goal of robotics research is to develop practical robots that can
act usefully in the real world. To that end, there has been recent progress in
bipedal locomotion research, which aims to allow robots to navigate
environments built for humans. There are few practical applications, however,
that can be solved with just locomotion capabilities. In particular, it is
important for robots to be able to handle and carry a variety of loads while
maintaining robust locomotion. Ideally, such load-carrying capabilities would
not require additional sensing, allowing for easily swapping loads without
needing additional hardware or wiring. Surprisingly, the problem of locomotion
control while carrying such loads is under-researched, especially for bipedal
robots.

In this work, we are particularly interested in dynamic loads, such as an
attached cart or container of liquid, rather than simple static loads like
rigidly attached fixed masses. Such loads can have their own internal dynamics
separate from the robot that can have effects even if the robot is stationary.
Thus, locomotion with dynamic loads requires not only being robust to
disturbances in the robot’s base dynamics, but also controlling the external
object via the locomotion gait. These kinds of scenarios have additional
underconstrained degrees of freedom that make the problem of locomotion much
more complicated. This raises the question of the effectiveness of recent
learning-based approaches to bipedal control when extended to handle dynamic
loads, especially without additional load-sensing capabilities.

Figure 1: Cassie with two 5 kg swinging masses attached. We are able to train
policies capable of walking with such a dynamic load and transfer simulation
trained policies to hardware.

One strategy for incorporating dynamic loads could be to use prior RL-trained
controllers (for unloaded settings) to walk very slowly with simple gaits in
order to avoid disturbing the load and having it impact the overall dynamics.
Rather, our goal is to achieve similar levels of dynamic walking performance
as these prior controllers but now with loads. To this end, we investigate the
RL approach to learning locomotion policies for a number of qualitatively
different dynamic loads with no additional load sensing beyond proprioception.

Our primary contribution is successful sim-to-real experiments of RL policies
capable of handling varying dynamic loads on the bipedal robot Cassie. We show
that simple straightforward training with specific loads is enough to result
in effective locomotion that generally improves over prior base controllers
trained without loads; no additional sensing capabilities, reward shaping, or
modification to the nominal learning setup is required. We provide an analysis
of the load-specific gaits compared to the base gaits to understand the key
strategic differences. In addition, we find that a general load policy trained
on all of the loads is able to reach nearly the same walking performance as
the load-specific policies. We also show that training load policies can be
dramatically accelerated by starting with a pre-trained unloaded policy.
Finally, we evaluate sim-to-real performance of the learned load policies. We
show that the policies do successfully transfer to the real-world, however,
the sim-to-real gap is larger than in previous unloaded work, which suggests
interesting future research opportunities.

##  II Related Work

###  II-A RL for Bipedal Locomotion

There have been many recent successes of RL applied to bipedal locomotion.
Many of the methods have relied on some kind of expert data in the reward
function in order to properly inform the policy how to walk. This is usually
in the form of trajectories [1, 2] or motion capture data [3, 4, 5]. Other
works have moved away from the use of such data to instead use a principled
reward function that describes walking by alternating between rewarding swing
and stance of each foot [6]. This allows the reward for a walking gait to be
described through simple gait parameters like swing ratio and cycle time. The
most common method to transfer simulator-trained RL policies to hardware is
via dynamics randomization [7, 8, 9], where the policy is trained over a
distribution of simulation model parameters to improve robustness to
inaccuracies in the simulator. In this work, we use a reward structure and
dynamics randomization similar to prior sim-to-real demonstrations on the
Cassie robot [6].

###  II-B Locomotion with Loads

Though there are many studies in the field of loco-manipulation [10, 11, 12],
legged loco-manipulation has received limited attention. Previous research on
quadrupeds with an attached manipulation arm [13, 14, 15, 16] rely mainly on
the statically stable nature of having four legs and mainly focus on holding
static weights. For example, recent works [13, 14] optimize center of mass and
limb motions which are tracked with a whole body controller. However, this
requires the load to be fixed to the robot since the whole body controller
assumes that the load is just another rigid part of the robot.

Study of loco-manipulation in the context of bipedal robots has largely
separated out the locomotion and manipulation problem. For example, [17] and
[18] investigate the problem of walking to an object, then picking it up or
moving it while stationary, not carrying a load while locomoting as we are
concerned with here. [19, 20] do target walking with a load, but use
statically stable walking gaits which limits the dynamic effects of the load
and results in slow walking speeds. [21] had a human scale DRC-Hubo robot
climb stairs while holding half-full 2 gallon buckets of water in each hand.
This was achieved by using an approximate model of the sloshing dynamics and a
PID controller feeding back on the joint angles, wave height of the container
liquid, and force data of the wrist, to adjust the arm joint angle. It is
noted that this method does use sensing not required for walking (wrist force
data and liquid wave height) and that the load’s force was “not enough to
change the center of gravity of the robot in an unstable manner,” so only
adjustment of the arm motion was required to be successful. In contrast, we
focus on the feasibility of learning to carry such dynamic loads without
specialized sensing and control architectures.

##  III Learning Load Policies

Our learning approach is based on a prior RL framework for training bipedal
locomotion policies [6] in an unloaded setting. Here we first summarize the
key elements of the framework, referring the reader to the original paper for
further details. Next, we describe our adaptations needed for extending to
locomotion under dynamic loads.

###  III-A Locomotion Learning Framework

Following prior work [6] we use an LSTM neural network as our primary
locomotion controller with two 128-dimensional recurrent hidden layers. The
network input contains: 1) the robot state information, including pelvis
orientation, rotational velocity, and joint positions and velocities, 2) a
clock signal consisting of a sine and cosine function with period length equal
to the desired cycle time, and 3) the desired forward velocity. The network
outputs position set points for each joint at 40Hz, which are passed to a PD
controller with fixed gains running at 2kHz. The network is trained in
simulation via the actor-critic PPO algorithm [22] using the clipped objective
along with gradient clipping. The simulation environment is based on the
cassie-mujoco-sim [23] library and the MuJoCo [24] physics engine. To
facilitate sim-to-real transfer, we use dynamics randomization in our training
where we randomize joint damping, joint mass, and ground friction following
prior work [6].

The reward signal for training is constructed the same as in [6] consisting of
the error between commanded velocity/orientation and actual
velocity/orientation, smoothing terms like pelvis acceleration penalty, torque
penalty, and an action difference cost, and foot costs to encourage
alternating swing and stance of each foot. However, to define the foot stance
and swing cost weightings, we replace the probabilistic function used in [6]
with a simple piecewise linear periodic function. In particular, our foot cost
weighting clock function is a set of anchor points defined by the swing ratio
and total cycle time, between which we linearly interpolate. Fig. 2 shows an
example linear clock with a swing ratio of 40% and a cycle time of 1 second.

Figure 2: An example of the piecewise-linear foot weighting clock with 40%
swing ratio and cycle time of 1 sec. The foot will be in swing for 0.4 sec,
20% of which is used to transition linearly between 0 and 1 and back again.
Weighting for stance is just 1 - swing weight. To get the weightings for the
right foot we just shift the left foot function forward half a cycle or 0.5
seconds.

Since, in this work, we focus on walking at different speeds, rather than
covering the entire gait spectrum as in prior work [6], we further simplify
the learning by only training on a fixed swing ratio and stepping frequency
for each speed. The phase offset gait parameter is always set to 0.5 for
symmetrical walking. We train on speeds from 0 m/s to 4 m/s and Table I shows
the heuristically defined swing ratio and stepping frequency for each speed.
For speeds 1 m/s to 3 m/s we linearly increase the swing ratio from 0.4 to 0.8
and stepping frequency from 1 Hz to 1.5 Hz in relation to the commanded speed.

Speed Range [m/s] | Swing Ratio [%] | Step Frequency [Hz]
---|---|---
0 - 1 | 40 | 1
1 - 3 | 40 - 80 | 1 - 1.5
3 - 4 | 80 | 1.5
TABLE I: Heuristics of gait parameters used in our running policy.

Using the above method for training in simulation without loads results in a
robust and stable walking policy that can accept velocity and orientation
commands. We will refer to this policy as the base policy, which will be used
as a point of comparison in our experiments and for bootstrapping learning of
load policies.

###  III-B Learning Locomotion for Dynamic Loads

Figure 3: All the different Cassie models that we experiment on, showing the 4
different dynamic loads as well as the base Cassie model.

Training policies to handle dynamic loads requires minimal modification to our
base training setup. The primary modification is contained to the simulator
where we define and use loaded robot models instead of the regular unloaded
model during simulations. The only exception is in cases where the reward
function might be modified to have a component related to a desirable load
property (see below), e.g. to punish spilling a liquid.

In this work, we consider 5 qualitatively different loaded Cassie robot
models, chosen to test different kinds of dynamic effects and possible use
cases.

  * •

Unloaded Cassie model, the regular Cassie model that has no load attached and
that the base walking policy is trained from.

  * •

Cassie Tray Box, which consists of a freely moving 5 kg box on top of a tray
attached to the top of the pelvis. The policy must learn to handle a free
external object such that it does not drop, similar to a waiter carrying a
cup.

  * •

Cassie Cart, which consists of a cart attached with ropes to the pelvis of
Cassie dragged along the ground. This load is used to test backwards pulling
forces.

  * •

Cassie Carrying Pole, which consists of a carrying pole attached to the pelvis
with hanging 5 kg weights on each end. Since the weights are free to swing in
every direction, this difficult load can easily cause both lateral and
torsional forces that induce unstable cycles and destroy locomotion.

  * •

Cassie Water Jug, which consists of a box rigidly attached to the pelvis with
water inside. To mimic the liquid dynamics without having to do fluid
simulations, we approximate the sloshing dynamics with a weight on springs in
every axis.

For each of these loads, expect for Tray Box, we used the same reward function
as described in Section II A that simply encourages smooth locomotion at a
desired velocity. For the Tray Box load, since there is the additional goal of
keeping the box on top of the tray, we add an extra reward component that
penalizes the distance between the position of the box and middle of the tray.
We additionally encourage the policy to keep the box on the tray by using an
extra termination condition. In addition to the reward and height termination
conditions described in [6] we terminate a trajectory if the box’s z height
falls below 50 cm, effectively meaning the box has fallen off the tray.

Finally, we experiment with three ways of training load policies. First, we
consider training load-specific policies from scratch starting from a random
neural network. Second, we consider training load-specific policies by
bootstrapping from the base policy, which simply initializes learning with the
high-performing policy trained without loads. This is similar to the type of
pre-training of general models in computer vision and natural language
processing, followed by fine-tuning for specific applications. Finally we
consider, training a general load policy that is trained on all load models
equally. This means that for each training iteration, the 50000 timesteps
sampled are split evenly between each of the 5 models, ensuring that the
policy sees equal amounts of experience from each load. Note that in this
case, since we cannot train the policy on multiple different reward functions,
we use the base walking reward function even for the Tray Box environment
unlike in the load specific policies. The “box falling off” termination
condition is kept for trajectories gathered in the tray box model. We train
both from scratch and bootstrapped versions of this general load policy.

##  IV Simulation Results

We consider the following metrics to evaluate walking performance under
dynamic loads in simulation.

Pass Rate. We randomly sample 5000 command sets, which consist of 2 random
speed commands and two random orientation commands, and measure how many sets
it can successfully follow. New commands are applied every 2.5 seconds. To
ensure some minimum change, the speed commands are randomly chosen between 0.5
and 2.0 m/s higher or lower than the previous command (within the command
limits of 0 to 4 m/s) and the orientation change command is similarly between
$30^{\circ}$ to $60^{\circ}$ left or right. A trial is considered a failure if
the robot falls down at any point.

Average Speed Error. For command set cases where the robot does not fall down,
we additionally track the average difference in desired and actual robot
velocity over the trajectory. We do not report this measure when the pass rate
is less than 50%.

Average Force. To evaluate push perturbation robustness, we perform a line
search in each direction to find the maximum force that the policy is able to
resist. The policy gets 3.0 seconds to recover from the 0.2 second impulse
that is applied to the robot pelvis. If the robot has not fallen down after
the 3 seconds, it is considered successful. We test every $10^{\circ}$ and
start from 50 N, increasing by 10 N until the force is too much and the falls
over. We then average over each direction to get a single number that gives us
an idea of how much force the policy can resist.

Max Speed. We test what is the maximum speed the policy is able to walk at
without falling over.

###  IV-A Base vs. Specialized Policies

In all load cases, we are able to successfully train specialized policies that
are able to walk with the load. Table II shows the walking performance for the
base walking policy evaluated on each different model, as well as the
performance of each load specific policy. Each policy was trained from
scratch. We can see how the performance of the base policy degrades when
evaluated on the dynamic loads it has never seen before, and how much
performance the load specific policies are typically able to gain back.

Pass Rate. In almost all cases the base policy fails on more command sets than
the load specific policies. This is especially true of the challenging Carry
Pole load, on which the base fails nearly immediately. Just by training on the
Carry Pole model, we are able to get an over 95% pass rate improvement, going
from falling down to stable and responsive walking.

Average Force Resistance. In all cases the load specific policies outperform
the base policy, being able to resist 10-20N more force on average. Again, we
see the largest improvement in the Carry Pole case, where force resistance
more than doubles. It is also interesting to note that some of the easier
loads actually have a stabilizing effect on the gait. For example, when
evaluated on the Cart load which is has never seen the before, the base
walking policy can actually resist 60 N more force on average than when
evaluated on the unloaded model. This is likely due to Cassie’s morphology,
particularly its lack of any arms or torso. This lack of inertia can make it
hard to resist twisting forces. Adding loads increases the overall inertia of
the robot and depending on the load can actually do so in a beneficial way
like with the cart load.

Average Speed Error. The load specific policies achieve lower average error
compared to the base policy on the Cart and Water Jug loads, which are the
only loads where the base policy has high pass rates and hence meaningful
speed errors. Interestingly, we see that the load specific policy for the
Water Jug achieves an error that is less than the base policy in the unloaded
condition. This is another example of unintended positive impacts of loaded
conditions.

|  |  Unloaded |  Tray Box | Cart |  Carry Pole |  Water Jug
---|---|---|---|---|---|---
Pass Rate | Base | .9784 |  .5364 | .9998 |  .0014 | .9746
Specific | .9784 |  .7112 | .996 |  .9578 | .976
General | .9992 |  .91 | 1.0 |  .8342 | .9998
Avg Force (N) | Base | 168.89 |  213.89 | 208.89 |  95 | 184.17
Specific | 168.89 |  236.39 | 218.89 |  221.11 | 202.22
General | 180.83 |  205.56 | 233.61 |  189.72 | 190.28
Avg Speed Error (m/s) | Base | .242 |  .152 | .3396 |  N/A | .257
Specific | .242 |  .175 | .3011 |  .3887 | .2353
General | .3182 |  .2915 | .4463 |  .437 | .3261
Max Speed (m/s) | Base | 3.9 |  1.8 | 3.5 |  N/A | 3.6
Specific | 3.9 |  2.1 | 2.6 |  2.5 | 3.0
General | 2.6 |  1.9 | 2.4 |  2.5 | 2.6
TABLE II: Walking performance comparison between the base policy, the
specialized policies, and the general load policy. The specialized policies
are only evaluated on the model they were trained on while the base and
general load policy are evaluated on every model. All policies are trained
from scratch.

Max Speed Something that was unexpected is that for the Cart and Water Jug
load the base policy has a higher max speed than the policy specifically
trained on the loads. This means that a higher walking speed is possible, but
for some reason a policy trained on the load won’t learn such a gait. We
hypothesize that this is because of the termination conditions which greatly
emphasizes staying alive over strictly following the speed command. This may
cause the load specific policies to be more conservative than the base policy,
which doesn’t see any load during training and likely triggers the termination
condition much less.

Though we are unable to gain back full performance of the base policy in some
cases, this is expected as adding in a load fundamentally changes the problem
and possibly changes the walking performance limit. It is unreasonable to
assume that a policy could walk as fast as before attaching swinging weights
to the robot.

###  IV-B Specialized vs. General Load Policies

We are also able to learn a general load policy that can handle a number of
different loads. It is reasonable to assume that the skills needed to handle
one load can be applied to a different load, and that similar sorts of
stabilizing, minimal movement gaits are needed no matter the load morphology.

Table II shows the walking performance of the general load policy versus each
model’s specifically trained policy. In most cases performance between the two
is comparable, though the general load policy does have higher speed error and
on the harder Tray Box and Carry Pole loads handles around 30 N less push
force. This indicates that a general load policy is viable, and that there is
perhaps some general load robustness skill that can be learned.

One case where the general load policy makes a significant compromise is max
speed for the unloaded condition. Since the policy has to handle walking for
all loads, the policy never gets the chance to explore higher speed walking as
it would cause failures on the load models. Thus even when evaluated in the
unloaded condition the policy can only reach 2.6 m/s. We also see a
significant decrease in pass rate for the Carry Pole load compared to the
specialized policy, which is arguable the most difficult dynamic load in our
set. While the general purpose load policy is much better than the base
policy, which has never experienced loads, this is a good example of when
specialized behavior is still beneficial. Finally, it is interesting that the
general policy typically improves the pass rate on other loads compared to the
base and specialized policies. This suggests that it has gained robustness due
to needing to handle a variety of dynamic conditions.

###  IV-C Learning from Scratch vs. Bootstrapping

We also investigate bootstrapping the learning from a base unloaded walking
policy compared to learning each load specific policy from scratch. Since in
every case the goal is still to walk forward, it makes sense that an expert
walking policy would have useful information and skills for the load policies,
and that we can quickly generalize from such an expert policy to learn such a
similar walking task.

We are able to learn bootstrapped policies for all loads without catastrophic
unlearning, even in the case of the carry pole load on which the base walking
policy completely fails. Even though the reward starts out drastically low as
the robot continually falls down, the base walking information embedded in the
policy network is still useful and the policy is quickly able to learn to walk
again. Fig. 4 shows the learning curves between the bootstrapped policies and
the ones learned from scratch. As we expected, the bootstrapped policies need
on the order of 100 million less samples to reach the same reward.

Figure 4: Learning curve comparison between learning from scratch and
bootstrapping from a successful walking policy for each load type. Note how in
all cases bootstrapping is significantly more sample efficient.

Table III shows that both types of policies have similar walking performance
and in some cases, even higher maximum speed and lower average speed error.
This means that we can bootstrap without worry of a loss in performance.

|  |  Tray Box |  Cart |  Carry Pole |  Water Jug
---|---|---|---|---|---
Pass Rate | Scratch |  .7112 |  .996 |  .9578 |  .976
Bootstrap |  .6944 |  .9986 |  .9834 |  .998
Avg Force (N) | Scratch |  236.39 |  218.89 |  221.11 |  202.22
Bootstrap |  239.44 |  189.44 |  226.67 |  192.78
Avg Speed Error (m/s) | Scratch |  .175 |  .3011 |  .3887 |  .2353
Bootstrap |  .1434 |  .2111 |  .3533 |  .2156
Max Speed (m/s) | Scratch |  2.1 |  2.6 |  2.5 |  3.0
Bootstrap |  2.0 |  3.2 |  2.4 |  3.1
TABLE III: Walking performance comparison between learning a policy from
scratch versus bootstrapping from an expert walking policy.

###  IV-D Gait Analysis

By comparing phase portraits between the base policy and each load specific
policy, we can examine what strategies the policy learns to deal with each
load.

For example, Fig. 5 compares the phase portraits of the tray box and base
policy in the top half. We show only the hip roll joints since it is the area
of main difference and the other actuated joints fall into similar cycles.
Note that the signs for the left and right hip are opposite because they are
on opposite sides of the pelvis; for the left roll negative angle means
towards the pelvis and for the right roll positive angle means towards the
pelvis. Here, the Tray Box policy in orange has noticeably more positive right
roll values and more negative left roll values. This indicates that the Tray
Box policy’s legs are kept closer towards the center of the pelvis. In other
words it’s taking narrower steps to help keep the pelvis more stable since a
narrower step path means the pelvis has to move less as the weight of the
robot shifts to be more under the foot currently in stance. A more stable
pelvis results in less movement in the box and the box not falling off the
tray.

Figure 5: Phase portrait comparison of the hip roll joints for the base policy
vs. the Tray Box policy and base policy vs. the Carry Pole policy. Note the
difference in walking strategy, as the tray box policy takes narrower steps
and the carry pole policy takes wider steps.

We can see that this is a different strategy from the Carry Pole policy. The
bottom half of Fig. 5 shows the phase portrait for the Carry Pole policy.
Looking again at the hip roll values, we see that the left values are more
positive and the right values are more negative. This indicates the exact
opposite strategy of the tray box policy: the Carry Pole policy takes wider
steps than normal instead of narrower. This is likely because the swinging
masses exert much larger forces on the robot than the loose box, and thus the
policy needs to walk with a wider stance in order to resist the extra lateral
and torsional forces applied by the swinging weights.

##  V Hardware Results

We test the tray box and carry pole policies on hardware in addition to the
base and general load policy. All policies are able to successfully transfer
to hardware. We refer readers to following link to view the hardware results:
https://youtu.be/IeSUM_ej8wE. We conduct our hardware tests by starting the
policy walking in place, then slowly increasing the speed until the robot
falls down or the policy can not make the robot walk any faster. The
successful policies are able to walk without falling down for over a minute
during our tests.

We note that while all policies are successful in that they can stably walk,
adding in dynamic loads significantly widens the sim-to-real gap. All of the
load policies reach significantly lower top speeds on hardware than in
simulation. While the base walking policy reaches a top speed of 3.0 m/s on
hardware, undershooting the simulation test by 0.9 m/s, the Tray Box policy
only reached 1.0 m/s, less than half of the simulation test’s 2.1 m/s. The
Carry Pole policy does even worse, reaching only 0.8 m/s compared to 2.5 m/s
in simulation.

We see this phenomenon in the general load policies as well. Though we see
little difference in behavior between the load specific and general policy for
the Tray Box load, on the more difficult Carry Pole load, the general load
trained from scratch failed where as the bootstrapped one was able to
successfully walk. This is likely because of all the extra noise that the from
scratch policy sees during training. The set of training states the from
scratch policy experiences is much more varied as it spends the entirety of
training seeing states from all loads. In contrast, the bootstrapped policy
spends over half of its experience only seeing states from the base model,
during the training of the initial base policy. The extra noise in the
training of the from scratch policy likely reduces its ability to transfer,
which is fine when the sim-to-real gap is small as for the unloaded and Tray
Box case, but when the sim-to-real gap widens like in the Carry Pole load, it
fails.

It is reasonable that adding dynamic loads increases the sim-to-real gap. More
moving bodies will result in more things to account for and more complex
dynamics to model, which all increases the margin of error in our simulation.
The deficiencies in hardware performance that we experience here indicate that
our (and RL’s most common) sim-to-real solution of dynamics randomization is
not enough for more dynamic tasks like carrying loads. This indicates that
further research into crossing the sim-to-real gap is necessary.

##  VI Conclusion

In this work we have shown that it is possible to train policies capable of
handling dynamic loads while locomoting for human scale bipedal robots. We
find that no additional sensing is required, and that the only modification
required from learning normal locomotion policies is to train in the context
of the desired loads. We also show that it is possible to learn a single
general load policy that can handle a number of different dynamic loads while
maintaining the walking performance of specialized policies trained on a
single specific load. In addition, we find that we can bootstrap from a
previously trained unloaded walking policy to quickly learn loaded walking
policies without loss of performance when compared to policies trained from
scratch. We demonstrate successful sim-to-real transfer for the load policies,
achieving a speed of 0.8 m/s while carrying two 5 kg swinging weights on each
side of the robot.

Our hardware results bring up interesting sim-to-real questions that could be
addressed in future work. Adding dynamic loads dramatically widens the sim-to-
real gap, and it seems like we are getting close to breaking the canonical
solution of dynamics randomization. It is likely that better strategies are
required in order to achieve simulation level of performance on hardware for
complex tasks like handling dynamic loads. More sophisticated sim-to-real
methods like shown in [9] are likely necessary to get full simulation
performance on hardware for loaded policies.

## Acknowledgments

We thank Carl Wilcox for design of hardware for physical load tests, Intel for
providing vLab resources and students at the Dynamic Robotics Laboratory for
helpful discussions.

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