unitree-g1/context/equations-and-bounds.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
equations-and-bounds	Equations, Constants, and Validation Bounds	established	reference/sources/official-product-page.md, reference/sources/official-developer-guide.md	[hardware-specs joint-configuration locomotion-control safety-limits push-recovery-balance whole-body-control motion-retargeting]	[forward_kinematics inverse_dynamics com zmp battery_runtime]	[dof actuator joint torque]	[]	[]	[Per-joint velocity limits (dq_max) — not published Exact link masses and inertia tensors (available in URDF) Ankle joint range of motion]

Equations, Constants, and Validation Bounds

Reference for all formulas, physical constants, and validation ranges used across the knowledge base. Always check values here before using a number in a calculation.

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1. Physical Constants

Constant	Symbol	Value	Unit
Gravity	g	9.81	m/s^2
Pi	pi	3.14159...	—

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2. Robot Parameters

Parameter	Symbol	Value	Unit	Source	Tier
Total mass	m_total	~35	kg	Official product page	T0
Standing height	h_stand	1.32	m	Official product page	T0
Folded height	h_fold	0.69	m	Official product page	T0
Width	w	0.45	m	Official product page	T0
Depth	d	0.20	m	Official product page	T0
Leg length (combined)	l_leg	0.60	m	Official product page	T0
Total DOF (base variant)	n_dof	23	count	Official spec sheet	T0
Total DOF (mid variant)	n_dof	29	count	Official spec sheet	T0
Total DOF (full variant)	n_dof	43	count	Official spec sheet	T0
Battery voltage (charger)	V_charge	54	V	Official spec sheet	T0
Battery voltage (nominal)	V_batt	~48.1	V	13-cell × 3.7V/cell	T3
Battery capacity	Q_batt	9,000	mAh	Official spec sheet	T0
Battery energy (estimated)	E_batt	~432	Wh	9Ah × 48.1V	T3
Max walking speed	v_max	2.0	m/s	Official spec sheet	T0
Max payload per arm (base)	m_pay	2	kg	Official spec sheet	T0
Max payload per arm (EDU)	m_pay	3	kg	Official spec sheet	T0
Max knee torque (base)	tau_knee	90	Nm	Official spec sheet	T0
Max knee torque (EDU)	tau_knee	120	Nm	Official spec sheet	T0
Control loop rate	f_ctrl	500	Hz	Developer guide	T0
DDS latency	t_dds	2	ms	Developer guide	T0
Runtime (continuous)	t_run	~2	hr	Official spec sheet	T0

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3. Joint Limits

Leg Joints (per leg)

Joint Name	q_min (deg)	q_max (deg)	q_min (rad)	q_max (rad)	tau_max (Nm)	Source	Tier
hip_yaw	-158	+158	-2.758	+2.758	—	Official specs	T0
hip_roll	-30	+170	-0.524	+2.967	—	Official specs	T0
hip_pitch	-154	+154	-2.688	+2.688	—	Official specs	T0
knee	0	+165	0	+2.880	90/120	Official specs	T0
ankle_pitch	—	—	—	—	—	Not documented	—
ankle_roll	—	—	—	—	—	Not documented	—

Waist Joints

Joint Name	q_min (deg)	q_max (deg)	q_min (rad)	q_max (rad)	Source	Tier
waist_yaw	-155	+155	-2.705	+2.705	Official specs	T0
waist_pitch	—	—	—	—	29/43-DOF only	—
waist_roll	—	—	—	—	29/43-DOF only	—

Arm/Hand Limits

Arm and hand joint limits not yet published in official documentation. Available in URDF files at unitree_ros/robots/g1_description/.

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4. Key Equations

4a. Forward Kinematics

T_ee = T_base * T_1(q_1) * T_2(q_2) * ... * T_n(q_n)

Where T_i(q_i) is the homogeneous transformation for joint i at angle q_i. DH parameters available in URDF model.

4b. Inverse Dynamics (Recursive Newton-Euler)

tau = M(q) * ddq + C(q, dq) * dq + G(q)

• M(q): mass/inertia matrix (29×29 for 29-DOF config)
• C(q, dq): Coriolis and centrifugal terms
• G(q): gravity vector (dominated by 35 kg total mass at g=9.81)

4c. Center of Mass

CoM = (1/m_total) * sum(m_i * p_i)

Where m_i and p_i are the mass and position of each link. Link masses available in URDF.

4d. Zero Moment Point (ZMP)

ZMP_x = (sum(m_i * (g * x_i - z_i * ddx_i + x_i * ddz_i))) / (sum(m_i * (g + ddz_i)))

Critical for bipedal balance — ZMP must remain within the support polygon. Note: The G1's RL-based controller learns ZMP-like constraints implicitly rather than using explicit ZMP computation. [T1]

4e. Battery Runtime Estimate

t_runtime = E_batt / P_avg

• E_batt ≈ 432 Wh [T3]
• P_avg varies: ~216 W for 2hr runtime (walking), lower for standing [T3]
• P_avg = E_batt / t_runtime = 432 / 2 ≈ 216 W average during continuous operation [T3]

4f. Control Loop Timing

dt_control = 1 / f_ctrl = 1 / 500 = 0.002 s = 2 ms

Motor commands should be published at this rate for smooth control. [T0]

4g. Support Polygon (Balance)

For double support (both feet on ground):
  Support polygon = convex hull of all foot contact points

For single support (one foot):
  Support polygon = convex hull of stance foot contact points

Balance condition (static):
  ZMP ∈ support polygon

The CoM projection onto the ground plane must remain within the support polygon for static stability. Dynamic stability (walking) allows temporary excursions. See push-recovery-balance. [T1]

4h. Control Barrier Function (CBF)

Safety function:    h(x) ≥ 0       (safe region)
CBF condition:      ḣ(x,u) + α·h(x) ≥ 0    (forward invariance)

• α > 0: convergence rate parameter
• For balance: h(x) = distance from CoM projection to nearest support polygon edge
• For joint limits: h(x) = min(q - q_min, q_max - q) See push-recovery-balance §3c and safety-limits §8. [T1]

4i. Kinematic Scaling (Motion Retargeting)

q_robot = retarget(p_human)

Simple scaling:
  p_robot_i = p_human_i * (l_robot_i / l_human_i)

Where:
  l_robot_i = robot limb segment length
  l_human_i = human limb segment length

After scaling, solve IK to get joint angles. Must clamp to robot joint limits. See motion-retargeting. [T1]

4j. Residual Policy Action

a_final = a_base + α * a_residual

Where:
  a_base = base controller output (stock or trained policy)
  a_residual = correction from residual policy
  α ∈ [0, 1] = safety scaling factor

Smaller α = more conservative (stay closer to base). See push-recovery-balance §3b. [T1]

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5. Validation Bounds

Use these to sanity-check calculations:

Quantity	Valid Range	Notes	Tier
Joint position (hip)	[-2.97, +2.97] rad	Most permissive hip range (roll)	T0
Joint position (knee)	[0, 2.88] rad	Extension only	T0
Joint torque (knee)	[-120, +120] Nm	EDU variant max	T0
Walking speed	[0, 2.0] m/s	Official max	T0
Battery voltage	~42-54 V	Depleted to fully charged estimate	T3
Control loop rate	500 Hz	Fixed by locomotion computer	T0
DDS latency	~2 ms	Internal network	T0
Robot mass	~35 kg	Without external payload	T0
Payload per arm	0-3 kg	EDU max	T0
Standing height	1.32 m	—	T0
Push recovery (standing)	~30-60 N·s	Impulse, estimated from papers	T3
CoM height (standing)	~0.7-0.9 m	Estimated from proportions	T3
Support polygon width	~0.15-0.25 m	Foot width, stance dependent	T3

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6. Unit Conversion Reference

From	To	Multiply by
degrees	radians	pi / 180
radians	degrees	180 / pi
rpm	rad/s	2*pi / 60
Wh	J	3600
mAh	Ah	1/1000
kg*m/s^2	N	1