---
id: equations-and-bounds
title: "Equations, Constants, and Validation Bounds"
status: established
source_sections: "reference/sources/official-product-page.md, reference/sources/official-developer-guide.md"
related_topics: [hardware-specs, joint-configuration, locomotion-control, safety-limits, push-recovery-balance, whole-body-control, motion-retargeting]
key_equations: [forward_kinematics, inverse_dynamics, com, zmp, battery_runtime]
key_terms: [dof, actuator, joint, torque]
images: []
examples: []
open_questions:
  - "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.

---

## 1. Physical Constants

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

---

## 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   |

---

## 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/`.

---

## 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]

---

## 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   |

---

## 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           |