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title

Denavit-Hartenberg Parameters

매 한 줄

"매 robot link 의 4 parameter 의 standard description". 매 robot manipulator 의 forward kinematics. 매 (a, α, d, θ) 의 의 매 link 의 transformation. 매 modern variant: 매 modified DH (Craig).

매 핵심

매 4 parameter (classical Denavit-Hartenberg)

a (link length): 매 z_{i-1} → z_i 의 X-axis 의 distance.
α (link twist): 매 z_{i-1} → z_i 의 angle.
d (link offset): 매 x_{i-1} → x_i 의 z-axis 의 distance.
θ (joint angle): 매 x_{i-1} → x_i 의 z-axis 의 rotation.

매 transformation matrix

T_i = Rot_z(θ) * Trans_z(d) * Trans_x(a) * Rot_x(α)

매 forward kinematics

매 each link 의 T_i 의 multiply.
매 base → end-effector 의 pose.

매 modified DH (Craig)

매 frame 의 link's proximal end 의 attach.
매 less ambiguity.

매 응용

Manipulator: 매 6-DOF arm.
Mobile robot: 매 articulated.
Surgical robot: 매 da Vinci.
Animation: 매 IK.
Drone arm: 매 aerial manipulation.

💻 패턴

Forward kinematics (Python)

import numpy as np

def dh_matrix(a, alpha, d, theta):
    ca, sa = np.cos(alpha), np.sin(alpha)
    ct, st = np.cos(theta), np.sin(theta)
    return np.array([
        [ct, -st*ca,  st*sa, a*ct],
        [st,  ct*ca, -ct*sa, a*st],
        [0,      sa,     ca,    d],
        [0,       0,      0,    1],
    ])

def forward_kinematics(dh_table, joint_angles):
    """dh_table: [(a, alpha, d, theta_offset), ...]"""
    T = np.eye(4)
    for (a, alpha, d, off), q in zip(dh_table, joint_angles):
        T = T @ dh_matrix(a, alpha, d, off + q)
    return T

# 매 example: PUMA 560
puma = [
    (0,       np.pi/2,  0,    0),
    (0.4318,  0,        0,    0),
    (0.0203, -np.pi/2,  0.15, 0),
    (0,       np.pi/2,  0.4318, 0),
    (0,      -np.pi/2,  0,    0),
    (0,       0,        0,    0),
]
T = forward_kinematics(puma, [0, np.pi/4, -np.pi/4, 0, np.pi/2, 0])
print(T[:3, 3])  # 매 end-effector position

Inverse kinematics (numerical Jacobian)

def jacobian(dh_table, q, eps=1e-6):
    n = len(q)
    p0 = forward_kinematics(dh_table, q)[:3, 3]
    J = np.zeros((3, n))
    for i in range(n):
        q1 = q.copy(); q1[i] += eps
        p1 = forward_kinematics(dh_table, q1)[:3, 3]
        J[:, i] = (p1 - p0) / eps
    return J

def ik_newton(dh_table, target, q0, max_iter=100, tol=1e-4):
    q = q0.copy()
    for _ in range(max_iter):
        p = forward_kinematics(dh_table, q)[:3, 3]
        err = target - p
        if np.linalg.norm(err) < tol: break
        J = jacobian(dh_table, q)
        dq = np.linalg.pinv(J) @ err
        q += dq
    return q

URDF integration

# URDF 의 DH 의 convert
import xml.etree.ElementTree as ET

def urdf_to_dh(urdf_path):
    """매 URDF joint 의 DH-style approx."""
    tree = ET.parse(urdf_path)
    dh = []
    for joint in tree.findall('joint'):
        if joint.attrib['type'] == 'revolute':
            origin = joint.find('origin')
            xyz = [float(x) for x in origin.attrib['xyz'].split()]
            rpy = [float(x) for x in origin.attrib['rpy'].split()]
            # 매 simplification — true DH extraction 의 nontrivial
            dh.append((xyz[0], rpy[0], xyz[2], 0))
    return dh

Workspace visualization

def workspace_sample(dh_table, joint_limits, n=5000):
    points = []
    for _ in range(n):
        q = [np.random.uniform(lo, hi) for lo, hi in joint_limits]
        p = forward_kinematics(dh_table, q)[:3, 3]
        points.append(p)
    return np.array(points)

# 매 plot
import matplotlib.pyplot as plt
pts = workspace_sample(puma, [(-np.pi, np.pi)] * 6)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(pts[:, 0], pts[:, 1], pts[:, 2], s=1)

Modified DH (Craig)

def mdh_matrix(a, alpha, d, theta):
    """매 frame at proximal end."""
    ca, sa = np.cos(alpha), np.sin(alpha)
    ct, st = np.cos(theta), np.sin(theta)
    return np.array([
        [ct,    -st,   0,   a],
        [st*ca,  ct*ca, -sa, -d*sa],
        [st*sa,  ct*sa,  ca,  d*ca],
        [0,      0,     0,    1],
    ])

매 결정 기준

상황	Approach
Standard manipulator	Classical DH
Avoiding singularity	Modified DH (Craig)
Modern simulation	URDF (rich features)
Closed-form IK	Pieper's solution (last 3 axes intersect)
Numerical IK	Jacobian-based
Beyond serial (parallel)	Stewart platform — DH X

기본값: 매 manipulator 의 DH + 매 forward kinematics + 매 numerical IK + 매 URDF for sim.

🔗 Graph

부모: Robotics
변형: Inverse-Kinematics
Adjacent: Degrees-of-Freedom

🤖 LLM 활용

언제: 매 robot manipulator design. 매 kinematics derivation. 매 sim setup. 언제 X: 매 parallel mechanism. 매 soft robot.

❌ 안티패턴

Confuse classical / modified: 매 transform 의 wrong.
Ignore singularity: 매 wrist 의 gimbal lock.
No joint limit: 매 unreachable.
Pure forward 의 trust: 매 IK 의 non-unique.

🧪 검증 / 중복

Verified (Spong/Hutchinson/Vidyasagar Robot Dynamics).
신뢰도 A.

🕓 Changelog

날짜	변경
2026-04-20	Auto-reinforced
2026-05-08	Phase 1
2026-05-10	Manual cleanup — DH parameter + 매 forward / IK / URDF / modified DH code

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