diff --git a/src/roboticstoolbox/ets/ET.py b/src/roboticstoolbox/ets/ET.py
index 0b00225a2..43824e96f 100644
--- a/src/roboticstoolbox/ets/ET.py
+++ b/src/roboticstoolbox/ets/ET.py
@@ -182,7 +182,7 @@ def Rx(
         """
         Pure rotation about the x-axis
 
-        :param param: rotation about the x-axis
+        :param param: rotation about the x-axis, or a joint name such as ``"q2"``, see :class:`~roboticstoolbox.ets._ET.BaseET`
         :param unit: angular unit, "rad" [default] or "deg"
         :param j: Explicit joint number within the robot
         :param flip: Joint moves in opposite direction
@@ -217,7 +217,7 @@ def Ry(
         """
         Pure rotation about the y-axis
 
-        :param param: rotation about the y-axis
+        :param param: rotation about the y-axis, or a joint name such as ``"q2"``, see :class:`~roboticstoolbox.ets._ET.BaseET`
         :param unit: angular unit, "rad" [default] or "deg"
         :param j: Explicit joint number within the robot
         :param flip: Joint moves in opposite direction
@@ -252,7 +252,7 @@ def Rz(
         """
         Pure rotation about the z-axis
 
-        :param param: rotation about the z-axis
+        :param param: rotation about the z-axis, or a joint name such as ``"q2"``, see :class:`~roboticstoolbox.ets._ET.BaseET`
         :param unit: angular unit, "rad" [default] or "deg"
         :param j: Explicit joint number within the robot
         :param flip: Joint moves in opposite direction
@@ -286,7 +286,7 @@ def tx(
         """
         Pure translation along the x-axis
 
-        :param param: translation distance along the x-axis
+        :param param: translation distance along the x-axis, or a joint name such as ``"q2"``, see :class:`~roboticstoolbox.ets._ET.BaseET`
         :param j: Explicit joint number within the robot
         :param flip: Joint moves in opposite direction
         :returns: An elementary transform
@@ -331,7 +331,7 @@ def ty(
         """
         Pure translation along the y-axis
 
-        :param param: translation distance along the y-axis
+        :param param: translation distance along the y-axis, or a joint name such as ``"q2"``, see :class:`~roboticstoolbox.ets._ET.BaseET`
         :param j: Explicit joint number within the robot
         :param flip: Joint moves in opposite direction
         :returns: An elementary transform
diff --git a/src/roboticstoolbox/ets/ET2.py b/src/roboticstoolbox/ets/ET2.py
index 983426310..4910d85e9 100644
--- a/src/roboticstoolbox/ets/ET2.py
+++ b/src/roboticstoolbox/ets/ET2.py
@@ -56,7 +56,7 @@ def R(
         """
         Pure rotation
 
-        :param param: rotation angle
+        :param param: rotation angle, or a joint name such as ``"q2"``, see :class:`~roboticstoolbox.ets._ET.BaseET`
         :param unit: angular unit, "rad" [default] or "deg"
         :param j: Explicit joint number within the robot
         :param flip: Joint moves in opposite direction
@@ -95,7 +95,7 @@ def tx(
         """
         Pure translation along the x-axis
 
-        :param param: translation distance along the x-axis
+        :param param: translation distance along the x-axis, or a joint name such as ``"q2"``, see :class:`~roboticstoolbox.ets._ET.BaseET`
         :param j: Explicit joint number within the robot
         :param flip: Joint moves in opposite direction
         :returns: An elementary transform
@@ -128,7 +128,7 @@ def ty(
         """
         Pure translation along the y-axis
 
-        :param param: translation distance along the y-axis
+        :param param: translation distance along the y-axis, or a joint name such as ``"q2"``, see :class:`~roboticstoolbox.ets._ET.BaseET`
         :param j: Explicit joint number within the robot
         :param flip: Joint moves in opposite direction
         :returns: An elementary transform
diff --git a/src/roboticstoolbox/ets/_ET.py b/src/roboticstoolbox/ets/_ET.py
index 420adf48e..8d0c55df2 100644
--- a/src/roboticstoolbox/ets/_ET.py
+++ b/src/roboticstoolbox/ets/_ET.py
@@ -52,28 +52,91 @@ def _resolve_param(
     return param
 
 
+#: characters that negate the joint variable when they start a joint descriptor
+_MINUS_SIGNS = ("-", "\u2212")  # hyphen-minus and the Unicode minus sign
+
+
 def _parse_joint_descriptor(s: str) -> tuple[int | None, bool]:
     """
     Parse a joint descriptor string into (jindex, flip).
 
-    A leading '-' sets flip (and is stripped); a leading '+' is stripped and
-    ignored. The first run of digits found anywhere in what remains becomes
-    the joint index - this treats 'theta2', 'q2', 'q(3)' and 'θ_3' the same
-    regardless of how the index is set off from the rest of the name. If no
-    digit is found, jindex is None, so the joint falls through to ETS's
-    existing auto-numbering for unassigned joints.
+    A leading '-' (or the Unicode minus sign) sets flip, and a leading '+' is
+    ignored; the sign is then stripped. The first run of digits found anywhere
+    in what remains becomes the joint index - this treats 'theta2', 'q2', 'q(3)'
+    and 'θ_3' the same regardless of how the index is set off from the rest of
+    the name. If no digit is found, jindex is None, so the joint falls through
+    to ETS's existing auto-numbering for unassigned joints.
+
+    Joint indices count from zero.  A name that ends with an underscore, such as
+    'A1_' or 'θ2_', counts from one: the index is one less than the digits.
+
+    :raises ValueError: the name counts from one and the number is zero
     """
-    flip = s.startswith("-")
-    if s[:1] in "+-":
+    flip = s[:1] in _MINUS_SIGNS
+    if s[:1] in _MINUS_SIGNS or s[:1] == "+":
         s = s[1:]
 
     match = re.search(r"\d+", s)
     jindex = int(match.group()) if match else None
 
+    if jindex is not None and s.endswith("_"):
+        if jindex == 0:
+            raise ValueError(
+                f"joint name {s!r} ends with an underscore, so the number must "
+                "count from one, but it is zero"
+            )
+        jindex -= 1
+
     return jindex, flip
 
 
+def _joint_display_name(s: str) -> str:
+    """
+    The name of a joint, as displayed, from its descriptor
+
+    The underscore that says the number counts from one is an instruction and is
+    not part of the name, so 'A1_' is displayed as 'A1'.  A name without a number
+    is shown as given.
+    """
+    body = s[1:] if s[:1] in _MINUS_SIGNS or s[:1] == "+" else s
+    if s.endswith("_") and re.search(r"\d", body):
+        return s[:-1]
+    return s
+
+
 class BaseET:
+    r"""
+    Elementary transform, the base class of :class:`ET` and :class:`ET2`
+
+    **Joint descriptors**
+
+    The parameter ``param`` of a transform may be a string naming a variable
+    joint, such as ``"q2"`` or ``"theta2"``, instead of a number.  The rules are:
+
+    - the first number in the string is the joint index, so ``"q0"`` is joint 0
+      and ``"theta2"`` is joint 2; ``"q(3)"`` and ``"θ_3"`` are joint 3
+    - a leading ``-`` (or the Unicode minus sign ``−``) negates the joint
+      variable, ``flip``, so ``"-q1"`` is joint 1, flipped.  A leading ``+`` is
+      ignored
+    - joint indices count from zero.  If the names count from one, as ``"A1"`` to
+      ``"A6"`` of a Kuka arm do, end the name with an underscore: ``"A1_"`` is
+      joint 0, and ``"-A1_"`` is joint 0, flipped.  The underscore is not part of
+      the name that is displayed
+    - a string with no number, such as ``"joint"``, is a variable joint that is
+      numbered automatically
+    - a string that is a number, such as ``"1.5"``, is a constant
+    - ``jindex`` and ``flip`` must not also be given
+
+    The name is shown when the transform is displayed, ``Rz(A1)``.
+
+    .. note:: A robot built from names ``"A1"`` to ``"A6"``, without the
+        underscores, has joint indices 1 to 6, and is rejected as the index 6 is
+        out of range for 6 joints.
+
+    .. versionchanged:: 1.5.0
+        A trailing underscore, and the Unicode minus sign, are accepted.
+    """
+
     def __init__(
         self,
         axis: str,
@@ -113,7 +176,7 @@ def __init__(
                         "joint descriptor for `param`"
                     )
                 jindex, flip = _parse_joint_descriptor(descriptor)
-                self._joint_name = descriptor
+                self._joint_name = _joint_display_name(descriptor)
                 param = None
 
         flip = bool(flip)  # None (not given, and not set above) -> False
diff --git a/src/roboticstoolbox/robot/BaseRobot.py b/src/roboticstoolbox/robot/BaseRobot.py
index 27c0aee67..cf4f8cddb 100644
--- a/src/roboticstoolbox/robot/BaseRobot.py
+++ b/src/roboticstoolbox/robot/BaseRobot.py
@@ -385,10 +385,20 @@ def visit_link(link: Any, jindex: Any) -> None:
             # Jindex set on all, check they are unique and contiguous
             if check_jindex:
                 jset = set(range(n))
+                jindices = {link.jindex for link in links if link.isjoint}
                 for link in links:
                     if link.isjoint and link.jindex not in jset:
+                        hint = ""
+                        if jindices == set(range(1, n + 1)):
+                            hint = (
+                                f"; joint indices count from zero, but these are"
+                                f" 1 to {n}.  Joint names that count from one,"
+                                " such as 'A1', need a trailing underscore,"
+                                " 'A1_', to be numbered from zero"
+                            )
                         raise ValueError(
-                            f"joint index {link.jindex} was repeated or out of range"
+                            f"joint index {link.jindex} was repeated or out of"
+                            f" range{hint}"
                         )
                     jset -= set([link.jindex])
                 if len(jset) > 0:  # pragma nocover  # is impossible
diff --git a/tests/test_et_joint_descriptor.py b/tests/test_et_joint_descriptor.py
new file mode 100644
index 000000000..e5eef47ee
--- /dev/null
+++ b/tests/test_et_joint_descriptor.py
@@ -0,0 +1,142 @@
+"""String joint descriptors for ET and ET2, such as "q2", "-theta3" and "A1_"
+
+A string for ``param`` that is not a number names a variable joint:
+
+- a leading ``-`` (or the Unicode minus) sets ``flip``, a leading ``+`` is ignored
+- the first run of digits is the joint index, counted from zero
+- a trailing underscore says the name counts from one, so the index is one less
+- the name is shown when the ET is displayed, without that underscore
+"""
+
+import numpy as np
+import numpy.testing as nt
+import pytest
+
+from roboticstoolbox import ET, ET2, ERobot, ETS
+
+MINUS = "−"  # Unicode minus sign
+
+# descriptor, jindex, flip, str(ET.Rz(descriptor))
+CASES = [
+    ("q0", 0, False, "Rz(q0)"),
+    ("q12", 12, False, "Rz(q12)"),
+    ("-q1", 1, True, "Rz(-q1)"),
+    ("+q1", 1, False, "Rz(+q1)"),
+    ("theta2", 2, False, "Rz(theta2)"),
+    ("θ_3", 3, False, "Rz(θ_3)"),
+    ("q(3)", 3, False, "Rz(q(3))"),
+    ("-q(3)", 3, True, "Rz(-q(3))"),
+    (f"{MINUS}q1", 1, True, f"Rz({MINUS}q1)"),
+    # one-based names, trailing underscore
+    ("A1_", 0, False, "Rz(A1)"),
+    ("A6_", 5, False, "Rz(A6)"),
+    ("A10_", 9, False, "Rz(A10)"),
+    ("A01_", 0, False, "Rz(A01)"),
+    ("-A1_", 0, True, "Rz(-A1)"),
+    (f"{MINUS}A2_", 1, True, f"Rz({MINUS}A2)"),
+    ("+A2_", 1, False, "Rz(+A2)"),
+    ("θ2_", 1, False, "Rz(θ2)"),
+    ("θ_3_", 2, False, "Rz(θ_3)"),
+    ("q(3)_", 2, False, "Rz(q(3))"),
+    # without the underscore the digits are the index, as written
+    ("A1", 1, False, "Rz(A1)"),
+    ("A6", 6, False, "Rz(A6)"),
+    # no digits: auto-numbered, and the underscore means nothing
+    ("joint", None, False, "Rz(joint)"),
+    ("joint_", None, False, "Rz(joint_)"),
+    ("-joint", None, True, "Rz(-joint)"),
+]
+
+
+@pytest.mark.parametrize("descriptor, jindex, flip, text", CASES)
+def test_descriptor_sets_index_flip_and_name(descriptor, jindex, flip, text) -> None:
+    et = ET.Rz(descriptor)
+    assert et.isjoint
+    assert et.jindex == jindex
+    assert et.isflip is flip
+    assert str(et) == text
+
+
+@pytest.mark.parametrize("make", [ET.Rx, ET.Ry, ET.Rz, ET.tx, ET.ty, ET.tz])
+def test_every_elementary_transform_takes_a_descriptor(make) -> None:
+    et = make("-A3_")
+    assert et.isjoint and et.jindex == 2 and et.isflip
+
+
+def test_planar_et2_takes_a_descriptor() -> None:
+    et = ET2.R("-A2_")
+    assert et.isjoint and et.jindex == 1 and et.isflip
+
+
+@pytest.mark.parametrize("descriptor", ["A0_", "-A0_", "q0_", "θ0_"])
+def test_one_based_name_cannot_be_zero(descriptor: str) -> None:
+    with pytest.raises(ValueError, match="count from one"):
+        ET.Rz(descriptor)
+
+
+@pytest.mark.parametrize("keyword", [{"jindex": 0}, {"flip": True}])
+def test_descriptor_and_keyword_are_exclusive(keyword) -> None:
+    with pytest.raises(ValueError, match="cannot specify"):
+        ET.Rz("A1_", **keyword)
+
+
+@pytest.mark.parametrize("value, expected", [("1.5", 1.5), ("-2", -2.0), ("1_0", 10.0)])
+def test_a_numeric_string_is_a_constant(value: str, expected: float) -> None:
+    et = ET.tz(value)
+    assert not et.isjoint
+    nt.assert_allclose(et.A()[2, 3], expected)
+
+
+def test_flipped_descriptor_negates_the_joint_variable() -> None:
+    q = 0.4
+    nt.assert_allclose(ET.Rz("-A1_").A(q), ET.Rz("A1_").A(-q), atol=1e-12)
+    nt.assert_allclose(ET.Rz("A1_").A(q), ET.Rz(jindex=0).A(q), atol=1e-12)
+
+
+def _kuka(names: list[str]) -> ETS:
+    # six revolute joints, alternating z and x axes, with link offsets
+    ets = ETS()
+    for k, name in enumerate(names):
+        axis = ET.Rz if k % 2 == 0 else ET.Rx
+        ets = ets * axis(name) * ET.tz(0.2 + 0.05 * k) * ET.tx(0.1)
+    return ets
+
+
+def test_one_based_names_build_the_same_robot_as_zero_based() -> None:
+    one = ETS(_kuka([f"A{i}_" for i in range(1, 7)]))
+    zero = ETS(_kuka([f"q{i}" for i in range(6)]))
+
+    assert [j.jindex for j in one.joints()] == [0, 1, 2, 3, 4, 5]
+    assert [str(j) for j in one.joints()][:2] == ["Rz(A1)", "Rx(A2)"]
+    assert "A1" in str(one) and "A6" in str(one) and "_" not in str(one)
+
+    r1, r0 = ERobot(one), ERobot(zero)
+    assert r1.n == r0.n == 6
+    q = np.array([0.1, -0.4, 0.7, 0.2, -0.9, 0.5])
+    nt.assert_allclose(r1.fkine(q).A, r0.fkine(q).A, atol=1e-12)
+    nt.assert_allclose(r1.jacob0(q), r0.jacob0(q), atol=1e-12)
+
+
+def test_flipped_one_based_joint_in_a_robot() -> None:
+    names = [f"A{i}_" for i in range(1, 7)]
+    plain = ERobot(ETS(_kuka(names)))
+    flipped = ERobot(ETS(_kuka(["-A1_"] + names[1:])))
+    q = np.array([0.3, 0.1, -0.2, 0.4, 0.5, -0.6])
+    q_flipped = q.copy()
+    q_flipped[0] = -q[0]
+    nt.assert_allclose(flipped.fkine(q).A, plain.fkine(q_flipped).A, atol=1e-12)
+
+
+def test_one_based_names_without_underscore_are_rejected_with_a_hint() -> None:
+    ets = ETS(_kuka([f"A{i}" for i in range(1, 7)]))
+    assert [j.jindex for j in ets.joints()] == [1, 2, 3, 4, 5, 6]
+    with pytest.raises(ValueError, match="trailing underscore") as info:
+        ERobot(ets)
+    assert "out of range" in str(info.value)
+
+
+def test_other_out_of_range_indices_get_no_hint() -> None:
+    ets = ETS(_kuka(["q0", "q1", "q2", "q3", "q4", "q9"]))
+    with pytest.raises(ValueError, match="out of range") as info:
+        ERobot(ets)
+    assert "underscore" not in str(info.value)