Current State of Legged Robots and Relation to
Current Research
Avik De
Cofounder & CTO, Ghost Robotics (Philadelphia)
Postdoc (Harvard SEAS)

Axes of confusion
Modularity
Modular
vs.
monolithic

Optimal control (and learning) vs. reactive control: Foresight
Optimal control
• (Discrete) construction of “value function”
• Given
• Infinite horizon cost
• find instantaneous “good” direction
• HJB equation
Reactive control
• Find a potential/energy-like function 𝑉(𝑥)
• “Myopic” energy-like
• “All-knowing” NF
• …
• ~simple to construct (which lends to easy
analysis)
• No built-in mechanism to account for the
“path cost” [cf. VV/MIT project]
[Zhong, …, Todorov (2013)]

Optimal control in practice (SOA)
• Give up hope on globally estimating VF—intractable
• Local estimates around “frequented” regions of state space
• “Trajectory optimization” -> local estimate around a specific state/input trajectory
• Analogies for RL
• Bad if perturbed to a different region of state space
• Online “re-estimation” of finite-horizon value function -> MPC
• Ensures the trajectory is “near” the current state

Analytical controller vs. Optimization: Implementation
Optimization
• Write as min
𝑢
… 𝑠. 𝑡. 𝑢 ∈ 𝑈
• Global solution if feasible space is convex
• In practice:
• QP of problem size 100—1000 < 1ms on desktop
processor
• Problem size 10—100 < 1ms on microcontroller
Analytical
• Inverse dynamics
• Use properties of mechanical system ->
natural control
• Challenges:
• Underactuation
• Constraints: 𝑢 ∈ 𝑈? Friction?
Recall (assuming known VF)
• Optimal control wants
• Reactive control wants 𝜕𝑉
𝜕𝑥
𝑓 𝑥, 𝑢 < 0

Modularity and Hierarchy in Optimal Control
Hover controller
𝑢𝑡
= ℎ(𝑥𝑡
) 𝑢𝑎
= 𝜋+ ∘ ℎ ∘ 𝜋(𝑥𝑎
)
𝑥𝑡
= 𝜋(𝑥𝑎
)
Hover value
function 𝑉𝑡
𝑥𝑡
𝑉
𝑎
= 𝑉𝑡
∘ 𝜋 𝑥𝑎
+ ganch
(xa
)
WIP

Fielded legged robotics: status update
Vision-60 (35kg, 12dof)
Research Scientific Commercial
Jerboa (3kg, 4dof)
Minitaur (6kg, 8dof direct drive)

Vision 60 design history
2018 2019
v1.0 v2.0
v3.0
Q1 Q1
v3.5
Prototype
Product
Customer
Q2 Q2
Q3 Q3
Q4 Q4

Design process => design science
• Power
consumption
• Gear ratio
selection
• Impact mitigation
• Reflected inertia
• Bus voltage
utilization
• Current @ motor
controller
• …
Power consumption (W)
Swing
Stance
Gait kinematics
“Simple models”
Past data
Gait dynamics
Forward
dynamics
Robot kinematic params Robot dynamic params Motor params

Efficiency: a fundamental constraint
*MIT cheetah is larger, “actuators only” mechanical
(not total) COT, carrying a small battery
[Seok et. al. (2015)]
Robot Cost of Transport
Vision 60 v3.5 0.80
Vision 60 v4.0 0.54
Spot Mini 0.91
MIT Cheetah 0.46*
Type Power (W)
Motors Mechanical 210
Low level electronics Electrical 20
Blind locomotion Algorithmic <1
Gait planner Algorithmic <5
Autonomy Algorithmic 15
Vision 60 v3.5 power budget

Modularity through hierarchy
?
?
Abstraction benefits
• Reduced (re)development
• Computational simplification
• Model-robustness

Dimension reduction and “reflexes” for perturbation rejection

Details of these reflexes make a huge difference in practice: outdoors
“details”
• Slope estimation
• Slip detection
and handling
• Stubbing
detection and
handling
• Early/late contact
handling
• “Re-swing”
reflexes
• …

Details of these reflexes make a huge difference in practice: indoors

Modularity through hierarchy
?
?
Posture control
Reflexes

Control as reduction (anchoring) + composition
[Full and Koditschek (1999)] “Templates and anchors…”
[De and Koditschek (IJRR 2018)]

Reduction (anchoring) exposes templates: reactive quadrupedal walking
Event-driven coupled
swing leg oscillators
WIP
Trot walk
(crawl-ish) walk

Can go further: templates are inevitable
• Subject to anchoring posture control,
• With sufficient actuated DOFs,
• (degrades gracefully with fewer)
• Reduced dynamics at least contain IP.
[De, Topping, Kod (in prep)]

Embedding pitch-steady target dynamics on floating torso models
• “Zero manifold” = pitch-steady locomotion (e.g. walking)
• Render ZM attracting and invariant
Limitations:
• Restriction dynamics are affected by anchoring force
• Form of restriction dynamics depends on virtual
constraint choice
Valid zero dynamics
[De, Topping, Kod (in prep)]
[Westervelt et. al (2007)]

Input-decoupled anchoring
• Can find reduced coordinates 𝑟 𝑞 s.t.
𝑢𝑎𝑛𝑐ℎ
does not appear in ሷ
𝑟
• 𝑟 𝑞 is ~ virtual leg pos
• SLIP dynamics are exactly embedded!
A new kind of anchoring
Input-decoupled anchoring with actuated
IP template behavior
Floating torso model 𝑥 ∈ 𝑆𝐸(2)
Conventional
anchoring
Input-
decoupled
anchoring
Invariant+attracting pitch-stable manifold (conventional anchoring/ZD)
[Full & Kod (1999)] [Westervelt et al (2007)]
[De, Topping, Kod (in prep)]

Application to open-loop control of leaping [De, Topping, Kod (in prep)]
• Use IP template behavior to design open-
loop leaping controllers
• Together with provably correct anchoring
• Application to leaping for mobile
manipulation (preliminary)

Modularity through hierarchy
?
Template control
Template dynamics
Posture control
Reflexes
“Blind” template behavior Sensor head
?

Combining reflexes with anticipatory planning (WIP)
• Receding horizon planning
• With template model
• Hierarchical dimension reduction enables real-
time solutions, robustness to model uncertainty

Conclusion
• Axes of confusion
• Modularity is a time-investment
that saves time, computational
effort, improves robustness
• Ghost is hiring ->
[email protected]
What do the animals do?